Review Article -Transform Method and Delta Type Fractional ...downloads.hindawi.com/journals/ddns/2015/852734.pdf · Review Article The Z -Transform Method and Delta Type Fractional

Review ArticleThe Z-Transform Method and Delta TypeFractional Difference Operators

Dorota Mozyrska and MaBgorzata Wyrwas

Faculty of Computer Science Bialystok University of Technology 15-351 Białystok Poland

Correspondence should be addressed to Małgorzata Wyrwas mwyrwaspbedupl

Received 11 December 2014 Accepted 26 January 2015

Academic Editor Delfim F M Torres

Copyright copy 2015 D Mozyrska and M WyrwasThis is an open access article distributed under the Creative CommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

TheCaputo- Riemann-Liouville- andGrunwald-Letnikov-type difference initial value problems for linear fractional-order systemsare discussed We take under our consideration the possible solutions via the classicalZ-transformmethod We stress the formulafor the image of the discrete Mittag-Leffler matrix function in theZ-transformWe also prove forms of images in theZ-transformof the expressed fractional difference summation and operators Additionally the stability problem of the considered systems isstudied

1 Introduction

Fractional operators are generalizations of the correspondingoperators of integer order as well in continuous as in discretecaseThe particular interpretation of the fractional derivativewas presented for example in [1] or [2] Fractional differ-ences used in models of control systems could be understandas (1) an approximation of continuous operators (see [3]) and(2) a possibility of involving some memory to differencesystems that is systems inwhich the current state depends onthe full history of systemsrsquo states Systems with fractionalderivatives and differences are widely discussed in manypapers However different authors could use various defini-tions of operators Some comparisons between three basictypes of fractional difference operators were studied in [4]and in [5] formultistep caseThere aremany papers in whichthe authors usually involve in discrete case the Grunwald-Letnikov-type fractional operator see for example [2 6ndash14]for case ℎ = 1 and also for general case ℎ gt 0 However theexact formulas for solutions are not used to considered linearinitial value problemsThe authors mainly use the recurrencemethod for stating their results Using the particular type ofdefinitions of fractional difference operators it was possible tostate the exact formulas for solutions to initial value problemswith the Riemann-Liouville- and the Caputo-type fractionaldifference operators The clue point is that solutions to initialvalue problems with the Grunwald-Letnikov-type operator

have the same values as those for systems with the Riemann-Liouville-type operator

Basic properties of fractional sums and operators weredeveloped firstly in [15] and continued by Atici and Eloe[16 17] Baleanu and Abdeljawad [18 19] Another concept ofthe fractional sumdifference was introduced in [20ndash22]

Here we attempt to review methods of solutions for frac-tional difference systems By comparing both types of solu-tions we produce the formula for theZ-transform of partic-ular type of the discreteMittag-Lefflermatrix functionThereare few papers where various kinds of discrete Mittag-Lefflerfunctions are discussed The important paper is [23] wherediscrete type and 119902-discrete analogues ofMittag-Leffler func-tion are presented Their relations to fractional differencesof the particular case are investigated The author considersalso applications of these functions to numerical analysis andintegrable systems However the formulas of the consideredfunctions mainly depend on the class of the type of calculusFor example in [19] the authors consider discrete Mittag-Leffler functions connected with nabla Caputo fractional lin-ear difference systems anduse there the nabla discrete Laplacetransform In [24] the authors consider a linear nabla (119902 ℎ)-difference equations of noninteger order In [25] the sequen-tial discrete fractional equations with some particular con-vention of the fractional difference are solved using the kindof fractional nabla discrete Mittag-Leffler functions

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2015 Article ID 852734 12 pageshttpdxdoiorg1011552015852734

2 Discrete Dynamics in Nature and Society

Table 1 BasicZ-transform formulas used in the text

119910 = 119910 (119899) Z[119910](119911)

120593120572(119899) = (

119899 + 120572 minus 1

119899) = (minus1)

119899

(minus120572

119899) (

119911

119911 minus 1)120572

(119886Δminus120572

ℎ119909)(119886 + 120572ℎ + 119899ℎ) (

ℎ119911

119911 minus 1)

120572

119883(119911)

120593lowast119896120572

(119899)1

119911119896(

119911

119911 minus 1)119896120572+1

(119886Δ120572

ℎlowast119909)(119886 + (1 minus 120572)ℎ + 119899ℎ) 119911 (

ℎ119911

119911 minus 1)

minus120572

119883(119911) minus 119911ℎminus120572

(119911

119911 minus 1)1minus120572

119909(119886)

119864(120572120573)

(120582 119899) =

infin

sum119896=0

120582119896

120593119896120572+120573

(119899 minus 119896) (119911

119911 minus 1)120573

(1 minus120582

119911(

119911

119911 minus 1)120572

)

minus1

120593119896120572

(119899)1

119911119896(

119911

119911 minus 1)(119896+1)120572

(119886Δ120572

ℎ119909)(119886 + (1 minus 120572)ℎ + 119899ℎ) 119911 (

ℎ119911

119911 minus 1)

minus120572

119883(119911) minus 119911ℎminus120572

119909(119886)

(0Δ120572

ℎ119910)(119899ℎ + ℎ) 119911 (

ℎ119911

119911 minus 1)

minus120572

119883(119911) minus 119911ℎminus120572

119909(119886)

There is the possible use of the generalized Laplacetransformation on time scales for that see for example[26] Such method was used in [16] and [27] under themethod called the 119877-transform and the Laplace transformrespectively for the situation where on the right side of thesystem there are functions that depend only on time Addi-tionally in the dissertation [28] the possible use of the Laplacetransform for semilinear systems ismentionedMore classicalsolution of the problem is also presented in [29] wherethe authors use the Z-transform to give the conditions forstability of systems with the Caputo-type operator Theconclusions were derived based on the image in the complexdomainThemain advantage of the use of theZ-transform isto introduce the natural language for discrete systems itmeans to work with sequences instead of discrete functionsdefined on various domains

The stability property is the main issue in dynamicalsystems As themost of descriptions of behaviour of processesare digital due to computing tools used at computers it isimportant to know some conditions for stability of discrete-time systemsThe results are similar for different operators In[30 31] the study of the stability problem for discrete frac-tional systems with the nabla Riemann-Liouville-type differ-ence operator is presented The authors stress that the Z-transform can be used as a very effective method for stabilityanalysis of systems In engineering problems more oftensystems are usedwith theGrunwald-Letnikov-type fractionaldifference One of the attempts to the stability of fractionaldifference systems is the notion of practical stability see [6 7]Since the linear systems are defined by some matrices theconditions connected with eigenvalues of these matrices arepresented for instance in [11 13 14]

Our paper is rather motivated by the idea of presentingimages via Z-transform of fractional difference operatorsand consequences into stability property We receive resultsfor stability of linear systems with three operators and with

steps ℎ gt 0 In the case of the Grunwald-Letnikov-type frac-tional operator our results are similar to those in [11 13 14]However steps in the proof of stability condition are based on[31]

The paper is organized as follows In Section 2 we presentthe preliminary material needed for further reading Solu-tions to the considered linear initial value problems with theCaputo-type fractional difference and the image of theunknown vector function are constructed in Section 3 As thesimple consequence of considering two types of solutionone from the classical approach and the second from theZ-transform we can write the images of the one-parameterMittag-Leffler functions Sections 4 and 5 deal with imagesof the unknown vector functions for linear initial valueproblems with the Riemann-Liouville- and the Grunwald-Letnikov-type fractional differences respectively Some com-parisons for theZ-transforms of Caputo-type and Riemann-Liouville-type operators are presented in Section 6 In Sec-tion 7 we study the stability of the considered systemsSection 8 provides the brief conclusions Additionally Table 1presents basicZ-transform formulas used in the text

2 Preliminaries

Now we recall the necessary definitions and technical propo-sitions that are used in the sequel therein the paper Let ℎ gt 0119886 isin R anddefine (ℎN)

119886= 119886 119886+ℎ 119886+2ℎ and120590(119905) = 119905+ℎ

for any 119905 isin (ℎN)119886 Hence 119905 = 119886 + 119899ℎ for 119899 isin N

0 For a

function 119909 (ℎN)119886

rarr R the forward ℎ-difference operator isdefined as (Δ

ℎ119909)(119905) = (119909(120590(119905))minus119909(119905))ℎ where 119905 isin (ℎN)

119886 see

[21 22] For an arbitrary 120572 isin R and 119905 isin R such that 119905ℎ isin

R minus1 minus2 the ℎ-factorial function is defined by 119905(120572)

ℎ=

ℎ120572(Γ((119905ℎ) + 1)Γ((119905ℎ) + 1 minus 120572)) where Γ is the Euler gammafunction see [21 22] We use the convention that division ata pole yields zero For ℎ = 1 we write shortly 119905

(120572) instead of119905(120572)

1and Δ instead of Δ

1

Discrete Dynamics in Nature and Society 3

Definition 1 (see [22]) For a function 119909 (ℎN)119886

rarr R thefractional ℎ-sum of order 120572 gt 0 is given by (

119886Δminus120572

ℎ119909)(119905) =

(1Γ(120572))sum119899

119904=0(119905 minus 120590(119886 + 119904ℎ))

(120572minus1)

ℎ119909(119886 + 119904ℎ)ℎ where 119905 = 119886 +

(120572 + 119899)ℎ 119899 isin N0and additionally (

119886Δ0

ℎ119909)(119905) = 119909(119905)

For 119886 = 0we write shortly Δminus120572ℎinstead of

0Δminus120572ℎ Note that

119886Δminus120572ℎ119909 (ℎN)

119886+120572ℎrarr R FromDefinition 1 it follows that for

119905 = 119886 + (120572 + 119899)ℎ

(119886Δminus120572

ℎ119909) (119905) = ℎ

120572

119899

sum119904=0

Γ (120572 + 119899 minus 119904)

Γ (120572) Γ (119899 minus 119904 + 1)119909 (119886 + 119904ℎ)

= ℎ120572

119899

sum119904=0

(119899 minus 119904 + 120572 minus 1

119899 minus 119904)119909 (119904)

(1)

where 119909(119904) = 119909(119886+119904ℎ) Observe that formula (1) has the formof the convolution of sequences Let us call for the momentthe family of binomial functions defined onZ parameterizedby 120583 isin R and given by values 120593

120583(119899) = ( 119899+120583minus1

119899) for 119899 isin N

0

and 120593120583(119899) = 0 for 119899 lt 0 Then using

120593120572(119899) = (

119899 + 120572 minus 1

119899) = (minus1)

119899

(minus120572

119899) (2)

we can write formula (1) in the form of the convolution of 120593120572

and 119909 In fact

(119886Δminus120572

ℎ119909) (119905) = ℎ

120572

(120593120572lowast 119909) (119899) (3)

where ldquolowastrdquo denotes a convolution operator that is (120593120572

lowast

119909)(119899) = sum119899

119904=0( 119899minus119904+120572minus1119899minus119904

) 119909(119904)We recall here that the Z-transform of a sequence

119910(119899)119899isinN0

is a complex function given by 119884(119911) = Z[119910](119911) =

suminfin

119896=0119910(119896)119911minus119896 where 119911 isin C is a complex number for which

this series converges absolutelyNote that Z[120593

120572](119911) = sum

infin

119896=0( 119896+120572minus1119896

) 119911minus119896 =

suminfin

119896=0(minus1)119896

( minus120572119896) 119911minus119896 = (119911(119911minus1))120572 In the stability analysis it is

important to prove to which classes of sequences 120593120572belongs

Let 119910(119899)119899isinN0

be a sequence and let 1199101

= suminfin

119904=0|119910(119904)| and

119910infin

= sup119899isinN0

|119910(119899)| be the norms defined in the space ofsequences Moreover let ℓ119901(N

0) 119901 isin 1infin be the spaces of

all sequences satisfying 119910119901lt infin

Lemma 2 For 120572 lt 0 and such that 120572 notin Zminusthe sequence 120593

120572

belongs to ℓ1(N0) that is suminfin

119904=0|120593120572(119904)| = 120593

1205721lt infin

Proof Let 120573 = minus120572 120573 gt 0 and [120573] = 119896 be the integer part of120573 that is 119896minus 1 lt 120573 le 119896 and 119896 isin N

0 Then for 119904 isin N

0such that

0 le 119904 le 119896we have ( 120573119904) gt 0 and for 119904 gt 119896we get (minus1)119904 ( 120573

119904) gt 0

Therefore

100381710038171003817100381712059312057210038171003817100381710038171 =

infin

sum119904=0

1003816100381610038161003816120593120572 (119904)1003816100381610038161003816 =

119896

sum119904=0

(120573

119904) +

infin

sum119904=119896+1

(minus1)119904

(120573

119904)

=

119896

sum119904=0

(120573

119904) +

infin

sum119904=0

(minus1)119904

(120573

119904) minus

119896

sum119904=0

(minus1)119904

(120573

119904)

= 2

[1198962]

sum119904=0

(120573

2119904 + 1) lt infin

(4)

Hence 120593120572isin ℓ1(N

0)

Proposition 3 For 120572 isin (0 1) one has 120593120572isin ℓinfin(N

0)

Proof Note that 120593120572(0) = 1 and

1003816100381610038161003816120593120572 (119899)1003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816

(minus120572) (minus120572 minus 1) sdot sdot (minus120572 minus (119899 minus 1))

119899

10038161003816100381610038161003816100381610038161003816

le1 sdot 2 sdot sdot 119899

119899= 1

(5)

for 119899 ge 1 Hence 120593120572infin

= 1 and consequently 120593120572isin ℓinfin(N

0)

that is 120593120572infin

= sup119899isinN0

|120593120572(119899)| lt infin

Proposition 4 For 119905 = 119886 + 120572ℎ + 119899ℎ isin (ℎZ)119886+120572ℎ

let one denote119910(119899) = (

119886Δminus120572

ℎ119909)(119905) and 119909(119899) = 119909(119886 + 119899ℎ) Then

Z [119910] (119911) = (ℎ119911

119911 minus 1)

120572

119883 (119911) (6)

where 119883(119911) = Z[119909](119911)

Proof In fact as we have from (3) that 119910(119899) = (119886Δminus120572

ℎ119909)(119905) =

ℎ120572(120593120572lowast 119909)(119899) then Z[119910](119911) = ℎ120572Z[120593

120572](119911)119883(119911) Moreover

Z[120593120572](119911) = (119911(119911 minus 1))120572 then we easily see equality

(6)

For ℎ = 1 (6) can be shortly written asZ[119886Δminus120572

119909](119911) =

(119911(119911minus1))120572119883(119911) where (119886Δminus120572

119909)(119886+120572+119899) = 119910(119899) is treatedas a sequence

Two fractional ℎ-sums can be composed as follows

Proposition 5 (see [32]) Let 119909 be a real-valued functiondefined on (ℎN)

119886 where 119886 ℎ gt 0 For 120572 120573 gt 0 the following

equalities hold (119886+120573ℎ

Δminus120572

ℎ(119886Δminus120573

ℎ119909))(119905) = (

119886Δminus(120572+120573)

ℎ119909)(119905) =

(119886+120572ℎ

Δminus120573

ℎ(119886Δminus120572

ℎ119909))(119905) where 119905 isin (ℎN)

119886+(120572+120573)ℎ

In [22] the authors prove the following lemma that givestransition between fractional summation operators for anyℎ gt 0 and ℎ = 1

Lemma 6 Let 119909 (ℎN)119886

rarr R and 120572 gt 0 Then(119886Δminus120572

ℎ119909)(119905) = ℎ120572(

119886ℎΔminus120572

1119909)(119905ℎ) where 119905 isin (ℎN)

119886+120572ℎand

119909(119904) = 119909(119904ℎ)

In our consideration the crucial role is played by thepower rule formula see [21] (

119886Δminus120572

ℎ120595)(119905) = (Γ(120583 + 1)Γ(120583 +

120572+1))(119905minus119886+120583ℎ)(120583+120572)

ℎ where120595(119903) = (119903minus119886+120583ℎ)

(120583)

ℎ 119903 isin (ℎN)

119886

119905 isin (ℎN)119886+120572ℎ

Then for 120595 equiv 1 we have 120583 = 0 Nextly taking119905 = 119899ℎ + 119886 + 120572ℎ 119899 isin N

0 we get (

119886Δminus120572

ℎ1)(119905) = (1Γ(120572 +

1))(119905 minus 119886)(120572)

ℎ= (Γ(119899 + 120572 + 1)Γ(120572 + 1)Γ(119899 + 1))ℎ120572 = ( 119899+120572

119899) ℎ120572

Particularly we have Z[119886Δminus120572ℎ1](119911) = ℎ120572(1 minus (1119911))120572+1 =

ℎ120572(119911(119911 minus 1))120572+1Now we adapt the above formulas to multivariable

fractional-order linear case and define Mittag-Leffler matrixfunctions In [19] the discrete Mittag-Leffler function isintroduced in the following way


Definition 7 Let 120572 120573 119911 isin C with Re(120572) gt 0 The discreteMittag-Leffler two-parameter function is defined as

E(120572120573)

(120582 119911)

=

infin

sum119896=0

120582119896 (119911 + (119896 minus 1) (120572 minus 1))

(119896120572)

(119911 + 119896 (120572 minus 1))(120573minus1)

Γ (120572119896 + 120573)(7)

We introduce the two-parameter Mittag-Leffler functiondefined in different manner and show that both definitionsgive the same values Moreover we use the function for thematrix case and prove by two ways its image in the Z-transform Let us define the following function

119864(120572120573)

(120582 119899) =

infin

sum119896=0

120582119896

120593119896120572+120573

(119899 minus 119896) (8)

In fact in the paper we use two of them namely119864(120572120572)

(120582 119899) =

suminfin

119896=0120582119896120593119896120572+120572

(119899 minus 119896) and 119864(120572)

(120582 119899) = 119864(1205721)

(120582 119899) =

suminfin

119896=0120582119896120593119896120572+1

(119899 minus 119896) For 120572 = 1 we have that 119864(11)

(120582 119899) =

119864(1)

(120582 119899) = (1 + 120582)119899

In our consideration an important tool is the image of119864(120572120573)

(120582 sdot) with respect to theZ-transform

Proposition 8 Let 119864(120572120573)

(120582 sdot) be defined by (8) Then

Z [119864(120572120573)

(120582 sdot)] (119911) = (119911

119911 minus 1)120573

(1 minus120582

119911(

119911

119911 minus 1)120572

)

minus1

(9)

where |119911| gt 1 and |119911 minus 1|120572|119911|1minus120572 gt |120582|

Proof By basic calculations we have

Z [119864(120572120573)

(120582 sdot)] (119911) =

infin

sum119896=0

infin

sum119899=0

120582119896

(119899 minus 119896 + 119896120572 + 120573 minus 1

119899 minus 119896) 119911minus119899

=

infin

sum119896=0

120582119896

119911minus119896

infin

sum119904=0

(119904 + 119896120572 + 120573 minus 1

119904) 119911minus119904

=

infin

sum119896=0

(120582

119911)

119896 infin

sum119904=0

(minus1)119904

(minus119896120572 minus 120573

119904) 119911minus119904

= (119911

119911 minus 1)120573 infin

sum119896=0

(120582

119911)

119896

(119911

119911 minus 1)119896120572

= (119911

119911 minus 1)120573

(1 minus120582

119911(

119911

119911 minus 1)120572

)

minus1

(10)

where the summation exists for |119911| gt 1 and |119911 minus 1|120572|119911|1minus120572 gt

|120582|

Proposition 9 Let 120572 isin (0 1] and 120573 lt 120572 + 1 Let 119877 be the setof all roots of the following equation

(119911 minus 1)120572

= 120582119911120572minus1

(11)

If all elements from 119877 are strictly inside the unit circle thenlim119899rarrinfin

119864(120572120573)

(120582 119899) = 0

Proof If all roots of (11) are strictly inside the unit circle thenusing theorem of final value for the Z-transform we easilyget the assertion

By Proposition 9 we get that for some order 120572 there is aldquogoodrdquo 120582 such that all elements from 119877 are inside the unitcircle

Corollary 10 Let 120582 isin R All elements from 119877 (Proposition 9)are inside the unit circle if and only if minus2

120572 lt 120582 lt 0

3 Caputo-Type Operator

In this section we recall the definition of the Caputo-typeoperator and consider initial value problems of fractional-order systems with this operator We discuss the problem ofsolvability of fractional-order systems defined by differenceequations with the Caputo-type operator In [33] versions ofsolutions of scalar fractional-order difference equation aregiven

Nextly we define the family of functions 120593lowast

119896120572 Z rarr R

parameterized by 119896 isin N0and by 120572 isin (0 1] with the following

values

120593lowast

119896120572(119899) =

(119899 minus 119896 + 119896120572

119899 minus 119896) for 119899 isin N

119896

0 for 119899 lt 119896

(12)

Proposition 11 Let 120593lowast119896120572

be the function defined by (12) Then

Z [120593lowast

119896120572] (119911) =

1

119911119896(

119911

119911 minus 1)119896120572+1

(13)

for 119911 such that |119911| gt 1

Proof This needs the following simple calculations

Z [120593lowast

119896120572] (119911)

=

infin

sum119899=0

120593lowast

119896120572(119899) 119911minus119899

=

infin

sum119899=119896

(119899 minus 119896 + 119896120572

119899 minus 119896) 119911minus119899

= 119911minus119896

infin

sum119904=0

(119904 + 119896120572

119904) 119911minus119904

= 119911minus119896

infin

sum119904=0

(minus1)119904

(minus119896120572 minus 1

119904) 119911minus119904

= 119911minus119896

(1 minus1

119911)minus119896120572minus1

=1

119911119896(

119911

119911 minus 1)119896120572+1

(14)

where |119911minus1| lt 1 to ensure the existence of the summation

Note that for any 120572 isin (0 1] 119896 119899 isin N0 the following

relations hold

(1) 120593lowast0120572

(119899) = 1 120593lowast119896120572

(119896) = 1 and 120593lowast1120572

(1) = 1(2) 120593lowast119896120572

(119899) = Γ(119899 minus 119896 + 119896120572 + 1)Γ(119896120572 + 1)Γ(119899 minus 119896 + 1) andas the division by pole gives zero the formula worksalso for 119899 lt 119896


(3) 120593lowast119896120572

(119899) = (1Γ(119896120572 + 1))(119899 minus 119896 + 119896120572)(119896120572) = ( 119899minus119896+119896120572119896120572

)

(4) Z[120593lowast

1120572](119911) = (1119911)(119911(119911 minus 1))

120572+1

= (1(119911 minus 1))(119911(119911 minus

1))120572

(5) Z[120593lowast119896120572

](119911) = (1119911119896) sdot (119911(119911 minus 1))119896120572+1 = (1(119911 minus 1)119896

) sdot

(119911(119911 minus 1))119896(120572minus1)+1 for 119896 isin N0

(6) For 120572 = 1 we have 120593lowast1198961

(119899) = ( 119899119896) and Δminus1 ( 119899minus1

119896) =

( 119899119896+1

)

According to item (6) in the properties of functions 120593lowast119896120572

for120572 = 1 we put some comment Note that Z [Δminus1 ( 119899

119896)] (119911) =

(119911(119911minus1)) sdotZ [( 119899119896)] (119911) = (119911(119911minus1)) sdot (1119911119896) sdot (119911(119911minus1))119896+1 =

1199112(119911 minus 1)119896+2 andZ [( 119899

119896+1)] (119911) = (1119911119896+1) sdot (119911(119911 minus 1))119896+2 =

119911(119911 minus 1)119896+2 ThereforeZ [Δminus1 ( 119899

119896)] (119911) = Z [( 119899

119896+1)] (119911) but

taking Z [Δminus1 ( 119899minus1119896

)] (119911) = (119911(119911 minus 1)) sdot Z [( 119899minus1119896

)] (119911) =

(119911(119911 minus 1)) sdot (1119911119896+1) sdot (119911(119911 minus 1))119896+1 = 119911(119911 minus 1)119896+2 so

Z [Δminus1 ( 119899minus1119896

)] (119911) = Z [( 119899119896+1

)] (119911)Taking particular value of 119911 = 119899 + 120572 minus 1 isin Z] and

considering the family of functions 120593lowast119896120572

the particular case oftheMittag-Lefler function fromDefinition 7 can be rewrittenas E(1205721)

(119860 119899 + ]) = suminfin

119896=0119860119896

120593lowast

119896120572(119899) = 119864

(120572)(119860 119899) where

] = 120572minus1 Note that in fact 119864(120572)

(119860 119899) = sum119899

119896=0119860119896120593lowast119896120572

(119899) so theright hand side is finite and consequently values 119864

(120572)(119860 119899)

always exist For 120572 = 1 there is the delta exponential function119864(1)

(119860 119899) = (119868 + 119860)119899

The properties of two-indexed functions 120593lowast119896119904

were givenin [34 Proposition 25] and generalized for 119899-indexed func-tions 120593

lowast

119896120572in [34] and in [35] with multiorder (120572) In the

paper therein we use functions 120593lowast119896120572

with step ℎ = 1 andone order The next proposition is the particular case for themulti-indexed functions

Proposition 12 (see [35]) Let 120572 isin (0 1] and ] = 120572 minus 1 Thenfor 119899 isin N

0one has

(Δminus120572

120593lowast

119896120572) (119899 + ]) = 120593

lowast

119896+1120572(119899) (15)

The next step is to present Caputo-type ℎ-differenceoperator

Definition 13 (see [32]) Let 120572 isin (0 1] The Caputo-typeℎ-difference operator

119886

Δ120572

ℎlowastof order 120572 for a function 119909

(ℎN)119886

rarr R is defined by

(119886

Δ120572

ℎlowast119909) (119905) = (

119886Δminus(1minus120572)

ℎ(Δℎ119909)) (119905) (16)

where 119905 isin (ℎN)119886+(1minus120572)ℎ

Note that119886

Δ120572

ℎlowast119909 (ℎN)

119886+(1minus120572)ℎrarr R Moreover for

120572 = 1 the Caputo-type ℎ-difference operator takes the form(119886

Δ1

ℎlowast119909)(119905) = (

119886Δ0

ℎ(Δℎ119909))(119905) = (Δ

ℎ119909)(119905) where 119905 isin (ℎN)

119886

Observe that using function 1205931minus120572

we can write the Caputo-type difference in the following way

(119886

Δ120572

ℎlowast119909) (119905) = ℎ

minus120572

(1205931minus120572

lowast Δ119909) (119899) (17)

where 119905 = 119886 + (1 minus 120572)ℎ + 119899ℎ and 119909(119899) = 119909(119886 + 119899ℎ)

For the Caputo-type fractional difference operator thereexists the inverse operator that is the tool in recurrence anddirect solving fractional difference equations

Proposition 14 (see [32]) Let 120572 isin (0 1] ℎ gt 0 119886 = (120572 minus 1)ℎand119909 be a real-valued function defined on (ℎN)

119886The following

formula holds (Δminus120572ℎ(119886

Δ120572

ℎlowast119909))(119905) = 119909(119905) minus 119909(119886) 119905 isin (ℎN)

120572ℎ

Similarly as in Lemma 6 there exists the transitionformula for the Caputo-type operator between the cases forany ℎ gt 0 and ℎ = 1

Lemma 15 (see [21]) Let 119909 (ℎN)119886

rarr R and 120572 gt 0 Then(119886

Δ120572

ℎlowast119909)(119905) = ℎminus120572(

119886ℎΔ1205721lowast

119909)(119905ℎ) where 119905 isin (ℎN)119886+120572ℎ

and119909(119904) = 119909(119904ℎ)

From this moment for the case ℎ = 1 we write119886ℎ

Δ120572

lowast=

119886ℎ

Δ120572

1lowastand similarly number 1 is omitted for two other

operators and fractional summations Then the result fromProposition 14 can be stated as

(Δminus120572

(119886ℎ

Δ120572

lowast119909)) (119899 + 120572) = 119909 (119899 + 120572) minus 119909 (119886) (18)

where 119899 isin N0 119909(119904) = 119909(119904ℎ) and 119886 = (120572 minus 1)ℎ

Proposition 16 For 119886 isin R 120572 isin (0 1] let one define 119910(119899) =

(119886

Δ120572

ℎlowast119909)(119905) where 119905 isin (ℎN)

119886+(1minus120572)ℎand 119905 = 119886+ (1minus120572)ℎ+ 119899ℎ

Then

Z [119910] (119911) = ℎminus120572

(119911

119911 minus 1)1minus120572

((119911 minus 1)119883 (119911) minus 119911119909 (119886)) (19)

where 119883(119911) = Z[119909](119911) and 119909(119899) = 119909(119886 + 119899ℎ)

Proof The equality follows from (119886

Δ120572

ℎlowast119909)(119905) = ℎminus120572(120593

1minus120572lowast

Δ119909)(119899)Then using the formula forZ[Δ119909](119911) = (119911minus1)119883(119911)minus

119911119909(0) = (119911 minus 1)119883(119911) minus 119911119909(119886) after simple calculations we get(19)

Observe that for 120572 = 1 we have Z[119910](119911) = (1ℎ)((119911 minus

1)119883(119911) minus 119911119909(0)) which agrees with the transform of dif-ference Δ

ℎof 119909 Using the Z-transform of the Caputo-type

operator for 120593lowast119896120572 we can calculate that for 119896 isin N

1the

following relation holds

(0Δ120572

lowast120593lowast

119896120572) (119899 + 1 minus 120572) = 120593

lowast

119896minus1120572(119899) (20)

while for 119896 = 0 it becomes zero that is (0Δ120572

lowast120593lowast0120572

)(119899+1minus120572) =

0Firstly let us consider systems with Caputo-type operator

for ℎ gt 0 given by the following form

(119886

Δ120572

ℎlowast119909) (119899ℎ) = 119860119909 (119899ℎ + 119886) 119899 isin N

0 (21)

where 119909 (ℎZ)119886

rarr R119901 120572 isin (0 1] 119886 = (120572minus1)ℎ and119860 is119901times119901

real matrix Let us consider the following initial condition119909(119886) = 119909

0isin R119901 for (21) System (21) with 119909(119886) = 119909

0can be

rewritten in the form

( ]Δ120572

lowast119909) (119899) = 119860119909 (119899 + ]) 119899 isin N

0 (22a)

119909 (]) = 1199090isin R119901

(22b)


where ] = 120572 minus 1 119909(119899 + ]) = 119909(119899ℎ + 119886) and 119860 = ℎ120572119860 We alsouse the notation 119909(119899) = 119909(119899 + ]) = 119909(119899ℎ + 119886) what helps toavoid the problemwith domains and allows to see the variablevector as a sequence

Proposition 17 (see [4]) Let 120572 isin (0 1] and 119886 = (120572 minus 1)ℎ Thelinear initial value problem

(119886

Δ120572

ℎlowast119909) (119905) = 119860119909 (119905 + 119886) 119905 isin (ℎN)

0(23a)

119909 (119886) = 1199090 1199090isin R119901 (23b)

has the unique solution given by the formula

119909 (119905) = E(120572)

(119860ℎ120572

119905

ℎ) 1199090= 119864(120572)

(119860ℎ120572

119899) 1199090

=

infin

sum119896=0

119860119896

ℎ119896120572

120593lowast

119896120572(119899) 1199090

(24)

where 119905 isin (ℎN)119886and 119905ℎ = 119899 + ] = 119899 + 120572 minus 1 isin N

120572minus1

The next proposition presents the proof of the Z-transform of the one-parameter Mittag-Leffler matrix func-tion where the idea of transforming a linear equation is used

Proposition 18 Let 120572 isin (0 1] and 119872(119899) = 119864(120572)

(119860ℎ120572

119899) =

suminfin

119896=0119860119896ℎ119896120572120593lowast

119896120572(119899) Then the Z-transform of the one-

parameter Mittag-Leffler function 119864(120572)

(119860ℎ120572 sdot) is givenby

Z [119872] (119911) =119911

119911 minus 1(119868 minus

1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

(25)

Proof Let us consider problem (22a)-(22b) and take the Z-transform on both sides of (22a) Then for ℎ = 1 from (19)we get (119911(119911 minus 1))1minus120572((119911 minus 1)119883(119911) minus 119911119909(119886)) = 119860ℎ120572119883(119911)where 119883(119911) = Z[119909](119911) and 119909(119899) = 119909(119886 + 119899ℎ) And aftercalculations we see that 119883(119911) = (119911(119911 minus 1))(119868 minus (1119911)(119911(119911 minus

1))120572119860ℎ120572)minus1119909

0 hence we get 119909(119905) = 119909(119899) = Zminus1[119883(119911)](119899) =

Zminus1[Z[119872](119911)](119899)1199090 that is the solution of the considered

initial value problem And then taking initial conditions asvectors from standard basis fromR119901 we get the fundamentalmatrix 119872 whose transform form should agree with thegiven

Let us consider the Caputo difference initial value prob-lem for the linear system that contains only one equationwith119860 = 120582ℎ

minus120572 that is (119886

Δ120572

ℎlowast119909)(119905) = 120582ℎminus120572119909(119905 + 119886) 119909(119886) = 1

Then119864(120572)

(120582 119899) = Zminus1[(119911(119911minus1))(1minus(120582119911)(119911(119911minus1))120572)minus1](119899) =

Zminus1[suminfin

119896=0(120582119896119911119896)(119911(119911 minus 1))119896120572+1](119899) is the scalar one-para-

meterMittag-Leffler function Observe that theZ-transformof this scalar Mittag-Leffler function can be obtained fromProposition 18 by substituting 119868 = 1 and 119860ℎ

120572 = 120582 into (25)

4 Riemann-Liouville-Type Operator

In this section we recall the definition of the Riemann-Liouville-type operator and consider initial value problemsof fractional-order systems with this operator Similarly as in

the case the Caputo-type operator we discuss the problem ofsolvability of fractional-order systems of difference equationsFamily of functions 120593

lowast

119896120572is useful for solving systems with

Caputo-type operatorWe also formulate the similar family offunctions that are used in solutions of systemswith Riemann-Liouville-type operator Let us define the family of functions120593119896120572

Z rarr R parameterized by 119896 isin N0and by 120572 isin (0 1]

with the following values

120593119896120572

(119899) =

(119899 minus 119896 + 119896120572 + 120572 minus 1

119899 minus 119896) for 119899 isin N

119896

0 for 119899 lt 119896

(26)

Proposition 19 Let 120593119896120572


Z [120593119896120572

] (119911) =1

119911119896(

119911

119911 minus 1)119896120572+120572

(27)

for 119911 such that |119911| gt 1

Proof By simple calculations we have

Z [120593119896120572

] (119911) =

infin

sum119899=0

120593119896120572

(119899) 119911minus119899

=

infin

sum119899=119896

(119899 minus 119896 + (119896 + 1) 120572 minus 1

119899 minus 119896) 119911minus119899

= 119911minus119896

infin

sum119904=0

(119904 + (119896 + 1) 120572 minus 1

119904) 119911minus119904

= 119911minus119896

infin

sum119904=0

(minus1)119904

(minus (119896 + 1) 120572

119904) 119911minus119904

= 119911minus119896

(1 minus1

119911)minus119896120572minus120572

=1

119911119896(

119911

119911 minus 1)(119896+1)120572

(28)

where the summation exists for |119911minus1| lt 1

Note that for any 120572 isin (0 1] and 119899 119896 isin N0the following

relations hold

(1) 1205930120572

(0) = 1 120593119896120572

(119896) = 1 and 1205931120572

(1) = 1(2) 120593119896120572

(119899) ge 0 and 120593lowast119896120572

(119899) ge 0

(3) 120593119896120572

(119899) = (119899 minus (119896 + 1) + (119896 + 1)120572)((119896+1)120572minus1)Γ((119896 + 1)120572)(4) Z[120593

0120572](119911) = (119911(119911 minus 1))120572Z[120593

1120572](119911) = (1119911)(119911(119911 minus

1))2120572(5) Z[120593

119896120572](119911) = (1119911119896)(119911(119911 minus 1))(119896+1)120572 for 119896 isin N

0

(6) For 120572 = 1 we have 1205931198961

(119899) = 120593lowast1198961

(119899) = ( 119899119896)

Taking particular value of 119911 = 119899 + 120572 minus 1 isin Z] andconsidering the family of functions 120593

119896120572the particular case

ofMittag-Leffler function fromDefinition 7 can be written asE(120572120572)

(119860 119899+]) = suminfin

119896=0119860119896120593119896120572

(119899) = 119864(120572120572)

(119860 119899) where ] = 120572minus

1 Note that in fact 119864(120572120572)

(119860 119899) = sum119899

119896=0119860119896120593119896120572

(119899) so the righthand side is finite and consequently values119864

(120572120572)(119860 119899) always


exist For 120572 = 1 there is a delta exponential function119864(11)

(119860 119899) = (119868 + 119860)119899

For the family of functions120593119896120572

we have similar behaviourfor fractional summations as for 120593lowast

119896120572so we can state the

following proposition

Proposition 20 Let 120572 isin (0 1] and ] = 120572minus1Then for 119899 isin N0

one has

(Δminus120572

120593119896120572

) (119899 + ]) = 120593119896+1120572

(119899) (29)

Proof In the proof we use the fact that Z[(Δminus120572120593119896120572

)(119899 +

])](119911) = (1119911)(119911(119911 minus 1))120572 sdot Z[120593119896120572

](119911) Then using item 5fromproperties of functions120593

119896120572 we see thatZ[(Δminus120572120593

119896120572)(119899+

])](119911) = (1119911119896+1)(119911(119911 minus 1))(119896+2)120572 = Z[120593119896+1120572

](119911) Hencetaking the inverse transform we get the assertion

The next presented operator is called the Riemann-Liouville-type fractional ℎ-difference operatorThe definitionof the operator can be found for example in [16] (for ℎ = 1)or in [21 22] (for any ℎ gt 0)

Definition 21 Let 120572 isin (0 1] The Riemann-Liouville-typefractional ℎ-difference operator

119886Δ120572

ℎof order 120572 for a function

119909 (ℎN)119886


(119886Δ120572

ℎ119909) (119905) = (Δ

ℎ(119886Δminus(1minus120572)

ℎ119909)) (119905) (30)


For 120572 isin (0 1] one can get

(119886

Δ120572

ℎlowast119909) (119905) = (

119886Δ120572

ℎ119909) (119905) minus

119909 (119886) sdot (119905 minus 119886)(minus120572)

ℎ

Γ (1 minus 120572)

= (119886Δ120572

ℎ119909) (119905) minus

119909 (119886)

ℎ120572(

119905 minus 119886

ℎminus120572

)

(31)

where 119905 isin (ℎN)119886 Moreover for 120572 = 1 one has (

119886

Δ1

ℎlowast119909)(119905) =

(119886Δ1

ℎ119909)(119905) = (Δ

ℎ119909)(119905)

The next propositions give useful identities of transform-ing fractional difference equations into fractional summa-tions

Proposition 22 (see [33]) Let 120572 isin (0 1] ℎ gt 0 119886 =

(120572 minus 1)ℎ and 119909 be a real-valued function defined on (ℎN)119886

The following formula holds (0Δminus120572

ℎ(119886Δ120572ℎ119909))(119905) = 119909(119905) minus 119909(119886) sdot

(ℎ1minus120572Γ(120572))119905(120572minus1)

ℎ= 119909(119905) minus 119909(119886) sdot ( 119905ℎ

120572minus1) 119905 isin (ℎN)

120572ℎ

In [21] the authors show that similarly as in Lemmas 6and 15 there exists the transition formula for the Riemann-Liouville-type operator between the cases for any ℎ gt 0 andℎ = 1

Lemma 23 Let 119909 (ℎN)119886

rarr R and 120572 gt 0 Then(119886Δ120572

ℎ119909)(119905) = ℎminus120572(

119886ℎΔ120572

1119909)(119905ℎ) where 119905 isin (ℎN)

119886+(1minus120572)ℎand

119909(119904) = 119909(119904ℎ)

For the case ℎ = 1 we write119886ℎ

Δ120572

=119886ℎ

Δ120572

1 Then the

result from Proposition 22 can be stated as

(Δminus120572

(119886ℎ

Δ120572

119909)) (119899 + 120572) = 119909 (119899 + 120572) minus 119909 (119886) sdot (119899 + 120572

119899 + 1)

(32)

where 119899 isin N 119909(119904) = 119909(119904ℎ) and with 119886 = (120572 minus 1)ℎ


(119886Δ120572

ℎ119909)(119905) where 119905 isin (ℎN)

119886+(1minus120572)ℎand 119905 = 119886 + (1 minus 120572)ℎ + 119899ℎ

Then

Z [119910] (119911) = 119911 (ℎ119911

119911 minus 1)

minus120572

119883(119911) minus 119911ℎminus120572

119909 (119886) (33)

where 119883(119911) = Z[119909](119911) and 119909(119899) = 119909(119886 + 119899ℎ)

Proof Let ℎ = 1 Then the assertion for ℎ gt 0 comes fromLemma 23 Let 119891(119899) = ( ]Δ

minus(1minus120572)

119909)(119899) Then Z[119910](119911) =

Z[Δ119891][119911] = (119911 minus 1)119865(119911) minus 119911119891(0) where 119865(119911) = Z[119891](119911) =

(119911(119911 minus 1))1minus120572119883(119911) and 119891(0) = 119909(119886) = 119909(0) Then we easilysee equality (33)

For 120572 = 1 we have the same as for the Caputo-typeoperator Z[119910](119911) = (1ℎ)((119911 minus 1)119883(119911) minus 119911119909(0)) which alsoagrees with the transform of difference Δ

ℎof 119909

Using the Z-transform of the Riemann-Liouville-typeoperator for 119896 isin N

1we can calculate that

(0Δ120572

120593119896120572

) (119899 + 1 minus 120572) = 120593119896minus1120572

(119899) (34)

while for 119896 = 0 we get 1205930120572

(119899) = ( 119899+120572minus1119899

) = 120593120572(119899) and

consequently (0Δ120572

1205930120572

)(119899+1minus120572) = (0Δ120572

120593120572)(119899+1minus120572) = 0

Now let us consider systems with the Riemann-Liouville-type operator for ℎ gt 0 given by the following form

(119886Δ120572

ℎ119909) (119899ℎ) = 119860119909 (119899ℎ + 119886) 119899 isin N

0 (35)

where 119909 (ℎZ)119886

rarr R119901 120572 isin (0 1] 119886 = (120572 minus 1)ℎ and 119860 is119901 times 119901 real matrix and with initial condition 119909(119886) = 119909

0isin R119901

System (35) with 119909(119886) = 1199090can be rewritten as follows

( ]Δ120572

119909) (119899) = 119860119909 (119899 + ]) 119899 isin N0 (36a)

119909 (]) = 1199090isin R119901

(36b)

where ] = 120572 minus 1 and 119909(119899 + ]) = 119909(119899ℎ + 119886) 119860 = ℎ120572119860 We alsouse the notation 119909(119899) = 119909(119899 + ]) = 119909(119899ℎ + 119886) what helps toavoid the problemwith domains and allows to see the variablevector as a sequence

Proposition 25 (see [4]) Let 120572 isin (0 1] and 119886 = (120572minus 1)ℎ Thelinear initial value problem

(119886Δ120572

ℎ119909) (119905) = 119860119909 (119905 + 119886) 119905 isin (ℎN)

0 (37a)

119909 (119886) = 1199090 1199090isin R119901

(37b)



119909 (119905) = E(120572120572)

(119860ℎ120572

119905

ℎ) 1199090= 119864(120572120572)

(119860ℎ120572

119899) 1199090

=

infin

sum119896=0

119860119896

ℎ119896120572

120593119896120572

(119899) 1199090

(38)


120572minus1


(119860ℎ120572 119899)Then the Z-transform of the Mittag-Leffler function119864(120572120572)

(119860ℎ120572 sdot) is given by

Z [119872] (119911) = (119911

119911 minus 1)120572

(119868 minus1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

(39)

Proof Let us consider problem (36a)-(36b) and take the Z-transform on both sides of (36a) Then we get from (33)for ℎ = 1 119911(119911(119911 minus 1))

minus120572

119883(119911) = 119911119909(119886) + 119860119883(119911) where119883(119911) = Z[119909](119911) and 119909(119899) = 119909(119886 + 119899ℎ) = 119909(] + 119899) And aftercalculations we see that 119883(119911) = (119911(119911 minus 1))120572(119868 minus (1119911)(119911(119911 minus

1))120572119860ℎ120572)minus11199090 hence 119909(119905) = 119909(119899) = Zminus1[119883(119911)](119899) =

Zminus1[Z[119872](119911)](119899) sdot1199090is the solution of the considered initial

value problem And then taking initial conditions as vectorsfrom standard basis from R119901 we get the fundamental matrix119872 whose transformZ[119872] is given by (39)

Let 119909 (ℎN)119886

rarr R Then for 119860 = 120582ℎminus120572 we have

(119886Δ120572

ℎ119909)(119905) = 120582ℎminus120572119909(119905 + 119886) 119909(119886) = 1 Taking into account the

image of the scalar Mittag-Leffler function in the Z-trans-form we get 119864

(120572120572)(120582 119899) = Zminus1[(119911(119911 minus 1))120572(1 minus (1119911)(119911(119911 minus

1))120572120582)minus1](119899) where 119868 = 1 and 119860ℎ120572 = 120582 are substituted into(39) Hence using the power series expansion the scalarMittag-Leffler function 119864

(120572120572)(120582 sdot) can be written as

119864(120572120572)

(120582 119899) = Zminus1[suminfin

119896=0((120582119896119911119896)(119911(119911 minus 1))119896120572+120572)](119899)

5 Gruumlnwald-Letnikov-Type Operator

The third type of the operator which we take under ourconsideration is the Grunwald-Letnikov-type fractional ℎ-difference operator see for example [2 6ndash14] for cases ℎ = 1

and also for general case ℎ gt 0

Definition 27 Let 120572 isin R The Grunwald-Letnikov-type ℎ-difference operator

119886Δ120572

ℎof order 120572 for a function 119909 (ℎN)

119886rarr

R is defined by

(119886Δ120572

ℎ119909) (119905) =

(119905minus119886)ℎ

sum119904=0

119886(120572)

119904119909 (119905 minus 119904ℎ) (40)

where

119886(120572)

119904= (minus1)

119904

(120572

119904)(

1

ℎ120572) with

(120572

119904) =

1 for 119904 = 0

120572 (120572 minus 1) sdot sdot sdot (120572 minus 119904 + 1)

119904 for 119904 isin N

(41)


(119886Δ120572

ℎ119909)(119905) where 119905 isin (ℎN)

119886and 119905 = 119886 + 119899ℎ 119899 isin N

0 Then

Z [119910] (119911) = (ℎ119911

119911 minus 1)

minus120572

119883 (119911) (42)

where 119883(119911) = Z[119909](119911) and 119909(119899) = 119909(119886 + 119899ℎ)

Proof Note thatZ[119910](119911) = Z[119886Δ120572

ℎ119909](119911) = Z[119886(120572)

119904lowast119909][119911] =

Z[119886(120572)119904

] sdot 119883(119911) = (ℎ119911(119911 minus 1))minus120572119883(119911)

The following comparison has been proven in [4]

Proposition 29 Let 119886 = (120572minus 1)ℎ Then nablaℎ(119886Δminus(1minus120572)

ℎ119909)(119899ℎ) =

(0Δ120572

ℎ119910)(119899ℎ) where 119910(119899ℎ) = 119909(119899ℎ + 119886) for 119899 isin N

0 or 119909(119905) =

119910(119905 minus 119886) for 119905 isin (ℎN)119886and (nabla

ℎ119909)(119899ℎ) = (119909(119899ℎ) minus 119909(119899ℎ minus ℎ))ℎ

Proposition 30 Let 119886 = (120572 minus 1)ℎ Then (0Δ120572

ℎ119910)(119905 + ℎ) =

(119886Δ120572

ℎ119909)(119905) where 119909(119905) = 119910(119905 minus 119886) for 119905 isin (ℎN)

119886

Proof Let119891(119899) = (0Δ120572

ℎ119910)(119899ℎ+ℎ)ThenbyProposition 28we

getZ[119891](119911) = 119911(ℎ119911(119911minus1))minus120572119883(119911)minus119911(0Δ120572

ℎ119910)(0) = 119911(ℎ119911(119911minus

1))minus120572119883(119911)minus119911ℎminus120572119910(0) = 119911(ℎ119911(119911minus1))minus120572 sdot119883(119911)minus119911119911ℎminus120572119909(119886) =

Z[119892] where 119883(119911) = Z[119909](119911) 119909(119899) = 119910(119899ℎ) = 119909(119886 + 119899ℎ) and119892(119899) = (

119886Δ120572

ℎ119909)(119905) 119905 isin (ℎN)

119886+(1minus120572)ℎ Hence taking the inverse

transform we get the assertion

From Proposition 30 we have the possibility of stat-ing exact formulas for solutions of initial value problemswith the Grunwald-Letnikov-type difference operator bythe comparison with parallel problems with the Riemann-Liouville-type operator And we can use the same formulafor the Z-transform of 119891(119899) = (

0Δ120572

ℎ119910)(119899ℎ + ℎ) as for the

Riemann-Liouville-type operator Then many things couldbe easily done for systems with the Grunwald-Letnikov-typedifference operator

The next proposition is proved in [36] for the case ℎ = 1

Proposition 31 The linear initial value problem

(0Δ120572

119910) (119905 + ℎ) = 119860119910 (119905) 119905 isin (ℎN)0

119910 (0) = 1199100 1199100isin R119901

(43)

has the unique solution given by the formula119910(119905) = Φ(119905)1199100for

any 119905 isin (ℎN)0and 119899 times 119899 dimensional state transition matrices

Φ(119905) are determined by the recurrence formula Φ(119905 + ℎ) =

(119860ℎ120572 + 119868120572)Φ(119905) with Φ(0) = 119868

Taking into account Propositions 25 and 30 we can statethe following result


(0Δ120572

ℎ119910) (119905 + ℎ) = 119860119910 (119905) 119905 isin (ℎN)

0

119910 (0) = 1199100 1199100isin R119901

(44)


119910 (119905) = 119864(120572120572)

(119860ℎ120572

119905

ℎ) 1199100 (45)

where 119905 isin (ℎN)0


As a simple conclusion we have that

Φ (119899ℎ) = Zminus1

[(119911

119911 minus 1)120572

(119868 minus1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

] (119899)

(46)

6 Comparisons and Examples

Here we present some comparisons for the Z-transforms ofCaputo- and Riemann-Liouville-type operators as the thirdone used in systemswith the left shift gives the same solutionsas the Riemann-Liouville-type operator Then we present theexample withmixed operators in two equations In this chap-ter we also state the formula in the complex domain of thesolution to the initial value problem for semilinear systemslike for control systems

Proposition 33 Let 119910(119899) = ((119886Δ120572

ℎminus119886

Δ120572

ℎlowast)119909)(119905) where 119905 =

119886 + (1 minus 120572)ℎ + 119899ℎ Then

Z [119910] (119911) = ℎminus120572

119911((119911

119911 minus 1)1minus120572

minus 1)119909 (119886) (47)

Proof It is enough to compare Propositions 16 and 24We canalso equivalently take under consideration (31) in the realdomain and observe thatZ[119910](119911) = ℎ

minus120572Z [( 119899minus120572+1minus120572

)] (119911)119909(119886) =

ℎminus120572 (119911suminfin

119904=0(minus1)119904

( 120572minus1119904

) 119911minus119904 minus 119911) 119909(119886) = ℎminus120572119911((119911(119911 minus 1))1minus120572 minus

1)119909(119886)

Comparing the Z-transform of two types of the Mittag-Leffler functions we get the following relation

Proposition 34 Let 120572 isin (0 1] 119899 isin N0 Then

119864(120572120572)

(119860 119899) = (119864(120572)

(119860 sdot) lowast 120593120572minus1

) (119899) (48)

Proof In fact if we take the Z-transform of the right sidethen we have

Z [119864(120572)


] (119911)

= Z [119864(120572)

(119860 sdot)] (119911)Z [120593120572minus1

] (119911)

=119911

119911 minus 1(119868 minus

1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

(119911

119911 minus 1)120572minus1

= (119911

119911 minus 1)120572

(119868 minus1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

(49)

Using formula (39) we get Z[119864(120572)


](119911) =

Z[119864(120572120572)

(119860 sdot)](119911) and consequently we reach the asser-tion

In the following example we consider control systemswith the Caputo- and Riemann-Liouville-type operators Weget the same transfer function for initial condition 119909(119886) = 0

Example 35 Let 120572 isin (0 1] ℎ = 1 and 119886 = 120572 minus 1 Let usconsider linear initial value problems of the following forms

(a) for the Caputo-type operator

(119886Δ120572

lowast119909) (119899) = 119860119909 (119886 + 119899) + 119861119906 (119899)

119910 (119899) = 119862119909 (119886 + 119899) + 119863119906 (119899) (50)

(b) for the Riemann-Liouville-type operator

(119886Δ120572

119909) (119899) = 119860119909 (119886 + 119899) + 119861119906 (119899)

119910 (119899) = 119862119909 (119886 + 119899) + 119863119906 (119899)(51)

with initial condition 119909(119886) = 0 and constant matrices 119860 isin

R119901times119901 119861 isin R119901times119898 119862 isin R119902times119901 119863 isin R119902times119898We solve the problems using theZ-transform Since119909(119886) =

0 then applying the Z-transform to the first equations inboth systems we get the following equation (119911(119911minus1))

1minus120572

(119911minus

1)119883(119911) = 119860119883(119911) + 119861119880(119911) Then 119883(119911) = (1(119911 minus 1)) sdot

(119911(119911 minus 1))120572minus1 sdot (119868 minus (1(119911 minus 1))(119911(119911 minus 1))120572minus1119860)minus1119861119880(119911) =

(1119911) sdot (119911(119911 minus 1))120572 sdot (119868 minus (1119911)(119911(119911 minus 1))120572119860)minus1119861119880(119911) where119883 = Z[119909] 119909(119899) = 119909(119886+119899) and119880 = Z[119906] Consequently forthe output of the considered systems in the complex domainwe have the following formula119884(119885) = (1119911)(119911(119911minus1))

120572119862(119868minus

(1119911)(119911(119911minus1))120572119860)minus1119861119880(119911)+119863119880(119911) where119884 = Z[119910]Thenthe solution for the output in the real domain is as follows119910(119899) = 119862(119891lowast119906)(119899)+119863119906(119899) where119891(119899) = 119864

(120572120572)(119860 119899minus1)119861We

can easily check the value at the beginning that 119910(0) = 119863119906(0)as 119864(120572120572)

(minus1) = 0

7 Towards Stability Analysis

Let us introduce the stability notions for fractional differencesystems involving the Caputo- Riemann-Liouville- andGrunwald-Letnikov-type operators

Firstly let us recall that the constant vector 119909eq is an equi-librium point of fractional difference system (21) if and only if(119886Δ120572

lowast119909eq)(119905) = 119891(119905 119909eq) = 0 (and (

119886Δ120572119894 119909eq)(119905) = 119891(119905 119909eq)

in the case of the Riemann-Liouville difference systemsof the form (35)) for all 119905 isin (ℎN)

0

Assume that the trivial solution 119909 equiv 0 is an equilibriumpoint of fractional difference system (21) (or (35)) Notethat there is no loss of generality in doing so because anyequilibrium point can be shifted to the origin via a changeof variables

Definition 36 The equilibrium point 119909eq = 0 of (21) (or (35))is said to be

(a) stable if for each 120598 gt 0 there exists 120575 = 120575(120598) gt 0 suchthat 119909(119886) lt 120575 implies 119909(119886+119896ℎ) lt 120598 for all 119896 isin N

0

(b) attractive if there exists 120575 gt 0 such that 119909(119886) lt 120575

implies

lim119896rarrinfin

119909 (119886 + 119896ℎ) = 0 (52)

(iv) asymptotically stable if it is stable and attractive

The fractional difference systems (21) (35) are called sta-bleasymptotically stable if their equilibrium point 119909eq = 0 isstableasymptotically stable

Example 37 Let us consider systems (23a)-(23b) or (37a)-(37b) but with 119860 = 0 Then for system (23a)-(23b) we havethat 119909(119886 + 119899ℎ) = 120593

lowast

0120572(119899)1199090= 1199090 while for system (37a)-(37b)


02

01

0

minus01

minus02

50 100 150 200xvalues

n steps

(a) The graph of 119909

02

01

0

minus01

minus02

50 100 150 200yvalues

n steps

(b) The graph of 119910

y

x

02

02

01

01

0

0

minus01

minus01

minus02

minus02

(c) Phase trajectory

Figure 1 The solution of the Caputo-type fractional-order difference initial value problem for linear system (21) with the initial condition119909(0) = 02 119910(0) = 02 where the matrix119860 is given in Example 40 and 120572 = 08 ℎ = 05 In this case the roots of (53) are strictly inside the unitcircle

the solution satisfies the following lim119899rarrinfin

119909(119886 + 119899ℎ) =

lim119899rarrinfin

1205930120572

(119899)1199090= 0 Hence then systems with the Caputo-

type operator are stable and systems with the Riemann-Liouville operator are attractive hence asymptotically stable

Proposition 38 Let 119877 be the set of all roots of the equation

det(119868 minus1

119911(

119911

119911 minus 1)120572

119860ℎ120572

) = 0 (53)

Then the following items are satisfied(a) If all elements from 119877 are strictly inside the unit circle

then systems (21) and (35) are asymptotically stable

(b) If there is 119911 isin 119877 such that |119911| gt 1 both systems (21) and(35) are not stable

Proof The main tool in the proof is Proposition 9 and themethod of decomposition of the matrix of systems intoJordanrsquos blocks Some basic tool is similar to those used in[30] However in the proof we do not use the same toolsas in the cited paper Only the beginning could be readas similar The Z-transform of 119909(sdot) is 119883(119911) = (119911(119911 minus

1))120573(119868 minus (1119911)(119911(119911 minus 1))120572119860ℎ120572)minus11199090

= 119875minus1(119911(119911 minus 1))120573(119868 minus

(1119911)(119911(119911 minus 1))120572

119869ℎ120572

)minus1

1198751199090 where 120573 = 1 for systems with

the Caputo-type operator and 120573 = 120572 for systems with


the Riemann-Liouville-type operator Moreover the condi-tion from Proposition 9 that 120573 lt 120572 + 1 is satisfied Let119860 = 119875119869119875

minus1 where 119875 is invertible and 119869 = diag(1198691 119869

119904)

and 119869119897are Jordanrsquos blocks of order 119903

119897 119897 = 1 119904 Let 119896

119894isin N1

be algebraic multiplicities of eigenvalues 120582119894

isin Spec(119860) let119901119894isin N1be their geometric multiplicities 119894 = 1 119898 The

matrix 119908minus1(119869) = (119868 minus (1119911)(119911(119911 minus 1))120572119869ℎ120572)minus1 is a diagonalblock matrixThe number of blocks corresponding to 120582

119894is 119901119894

and their form is given by the upper triangular matrix

(

(

119908minus1 (120582119894) 119908minus2 (120582

119894) 119908minus119903119902 (120582

119894)

0 119908minus1 (120582119894) 119908minus119903119902+1 (120582

119894)

d

0 0 119908minus1 (120582119894)

)

)

(54)

where 119902 = 1 119901119894and 119903119902is the size of the block Because

det(119908(119860)) = det(119908(119869)) = prod119898

119894=1119908119896119894(120582

119894) and 119883(119911) is formed by

a linear combination of convolutions of 120593120573with the rational

functions from (54) 119877 is the set of all poles of (119911 minus 1)119883(119911) as120572 minus 120573 + 1 gt 0

(a) If all elements from119877 are strictly inside the unit circlethen lim

119899rarrinfin119909(119899) = lim

119911rarr1(119911 minus 1)119883(119911) = 0

(b) If there is an element from 119877 that |119911| gt 1 thenlim sup

119899rarrinfin|119909(119886 + 119899)| = infin and systems (21) and (35)

are not stable

Now we can easily derive some basic corollary

Corollary 39 If there is 120582 isin Spec(A) such that |120582| gt (2ℎ)120572then system (21) is not asymptotically stable Or equivalentlyif system (21) is asymptotically stable then |120582| le (2ℎ)

120572 for all120582 isin Spec(119860)

Example 40 Let us consider the family of planar systemswithorder 120572 isin (0 1] described by the matrix A = [ minus2

120572minus2120572

2120572minus2120572 ]

Observe that det 119860 = 22120572+1 = 0 so the considered planarsystems have the unique equilibrium point 119909

119890= 0 Since

condition (53) does not depend on the type of an operatorwe take into account systems (21) with the Caputo-typedifference operator In our case we get for (i) 120572 = 08 ℎ = 05

that elements from 119877 are inside the unit circle and the systemis stable trajectories are presented at Figure 1 but for (ii) 120572 =

05 ℎ = 05we get that the system is unstable see Figure 2 Inthis case we can have stability for smaller step ℎ For the givenmatrix 119860 and 120572 the stability of the systems depends also onthe length of the step It is easy to see that 120582

1= minus2120572 minus 2120572119894

1205822

= minus2120572 + 2120572119894 isin Spec(119860) and |120582119895| = 2120572+05 119895 = 1 2 Then

for ℎ = 05 we have (2ℎ)120572 = 22120572 and consequently if ℎ =

05 and 120572 lt 05 then by Corollary 39 system (21) is notasymptotically stable Additionally one can see that for 120572 =

05 and ℎ = 05 we get |120582119895| = (2ℎ)120572 = 2 and the equilibrium

point is not stable Therefore the condition |120582| le (2ℎ)120572

given for eigenvalues of the matrix describing the system isnot sufficient for asymptotic stability of the system

0

y

x

times107

times107

15

10

10

05

minus10

minus10

minus05

Figure 2 The solution of the Caputo-type fractional-order differ-ence initial value problem for linear system (21) with the initialcondition 119909(0) = 02 119910(0) = 02 where the matrix 119860 is given inExample 40 and 120572 = 05 ℎ = 05 In this case the roots of (53) areoutside the unit circle

8 Conclusions

The Caputo- Riemann-Liouville- and Grunwald-Letnikov-type fractional-order difference initial value problems forlinear systems are discussed We stress the formula for theZ-transform of the discrete Mittag-Leffler matrix functionAdditionally the stability problem of the considered systemsis studied The next natural step is to consider the semilinearcontrol systems and develop the controllability property viathe complex domain

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are very grateful to anonymous reviewers forhelping them to improve the paperTheprojectwas supportedby the founds ofNational Science Centre granted on the basesof the decision no DEC-201103BST703476 The workwas supported by Bialystok University of Technology GrantGWM32012

References

[1] J A TMachado ldquoAnalysis and design of fractional-order digitalcontrol systemsrdquo Systems Analysis Modelling Simulation vol 27no 2-3 pp 107ndash122 1997

[2] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999


[3] DMozyrska E Girejko andMWyrwas ldquoFractional nonlinearsystems with sequential operatorsrdquo Central European Journal ofPhysics vol 11 no 10 pp 1295ndash1303 2013

[4] D Mozyrska E Girejko and M Wyrwas ldquoComparison of h-difference fractional operatorsrdquo in Advances in the Theory andApplications of Non-Integer Order Systems W Mitkowski JKacprzyk and J Baranowski Eds vol 257 of Lecture Notes inElectrical Engineering pp 191ndash197 Springer Berlin Germany2013

[5] D Mozyrska ldquoMultiparameter fractional difference linear con-trol systemsrdquoDiscrete Dynamics inNature and Society vol 2014Article ID 183782 8 pages 2014

[6] M Busłowicz and T Kaczorek ldquoSimple conditions for practicalstability of positive fractional discrete-time linear systemsrdquoInternational Journal of Applied Mathematics and ComputerScience vol 19 no 2 pp 263ndash269 2009

[7] M Busłowicz ldquoRobust stability of positive discrete-time linearsystems of fractional orderrdquo Bulletin of the Polish Academy ofSciences Technical Sciences vol 58 no 4 pp 567ndash572 2010

[8] T Kaczorek ldquoFractional positive linear systemsrdquo Kybernetesvol 38 no 7-8 pp 1059ndash1078 2009

[9] T Kaczorek ldquoReachability of cone fractional continuous-timelinear systemsrdquo International Journal of Applied Mathematicsand Computer Science vol 19 no 1 pp 89ndash93 2009

[10] P Ostalczyk ldquoEquivalent descriptions of a discrete-timefractional-order linear system and its stability domainsrdquo Inter-national Journal of Applied Mathematics and Computer Sciencevol 22 no 3 pp 533ndash538 2012

[11] M Busłowicz and A Ruszewski ldquoNecessary and sufficient con-ditions for stability of fractional discrete-time linear state-spacesystemsrdquo Bulletin of the Polish Academy of Sciences TechnicalSciences vol 61 no 4 pp 779ndash786 2013

[12] A Dzielinski and D Sierociuk ldquoStability of discrete fractionalorder state-space systemsrdquo Journal of Vibration and Control vol14 no 9-10 pp 1543ndash1556 2008

[13] R Stanisławski and K J Latawiec ldquoStability analysis fordiscrete-time fractional-order LTI state-space systems Part INew necessary and sufficient conditions for the asymptoticstabilityrdquo Bulletin of the Polish Academy of Sciences TechnicalSciences vol 61 no 2 pp 353ndash361 2013

[14] R Stanisławski and K J Latawiec ldquoStability analysis fordiscrete-time fractional-order LTI state-space systems Part IInew stability criterion for FD-based systemsrdquo Bulletin of thePolish Academy of Sciences Technical Sciences vol 61 no 2 pp363ndash370 2013

[15] K S Miller and B Ross ldquoFractional difference calculusrdquo inProceedings of the International Symposium on Univalent Func-tions Fractional Calculus and Their Applications pp 139ndash152Nihon University Koriyama Japan May 1988

[16] F M Atıcı and PW Eloe ldquoA transformmethod in discrete frac-tional calculusrdquo International Journal of Difference Equationsvol 2 no 2 pp 165ndash176 2007

[17] F M Atici and P W Eloe ldquoInitial value problems in discretefractional calculusrdquo Proceedings of the American MathematicalSociety vol 137 no 3 pp 981ndash989 2009

[18] T Abdeljawad and D Baleanu ldquoFractional differences andintegration by partsrdquo Journal of Computational Analysis andApplications vol 13 no 3 pp 574ndash582 2011

[19] T Abdeljawad F Jarad and D Baleanu ldquoA semigroup-likeproperty for discrete Mittag-Leffler functionsrdquo Advances inDifference Equations vol 2012 article 72 2012

[20] N RO Bastos RAC Ferreira andD FMTorres ldquoNecessaryoptimality conditions for fractional difference problems of thecalculus of variationsrdquo Discrete and Continuous DynamicalSystems vol 29 no 2 pp 417ndash437 2011

[21] R A C Ferreira and D F Torres ldquoFractional h-differenceequations arising from the calculus of variationsrdquo ApplicableAnalysis andDiscreteMathematics vol 5 no 1 pp 110ndash121 2011

[22] N R O Bastos R A C Ferreira andD FM Torres ldquoDiscrete-time fractional variational problemsrdquo Signal Processing vol 91no 3 pp 513ndash524 2011

[23] ANagai ldquoDiscreteMittag-Leffler function and its applicationsrdquoDepartmental Bulletin Paper 2003

[24] J Cermak T Kisela and L Nechvatal ldquoDiscrete Mittag-Lefflerfunctions in linear fractional difference equationsrdquoAbstract andApplied Analysis vol 2011 Article ID 565067 21 pages 2011

[25] N Acar and F M Atıcı ldquoExponential functions of discrete frac-tional calculusrdquo Applicable Analysis and Discrete Mathematicsvol 7 no 2 pp 343ndash353 2013

[26] R AgarwalM Bohner DOrsquoRegan andA Peterson ldquoDynamicequations on time scales a surveyrdquo Journal of Computationaland Applied Mathematics vol 141 no 1-2 pp 1ndash26 2002

[27] M T Holm ldquoThe Laplace transform in discrete fractionalcalculusrdquo Computers amp Mathematics with Applications vol 62no 3 pp 1591ndash1601 2011

[28] M T Holm The theory of discrete fractional calculus develop-ment and application [PhD thesis] University of NebraskandashLincoln 2011

[29] R Abu-Saris and Q Al-Mdallal ldquoOn the asymptotic stabilityof linear system of fractional-order difference equationsrdquo Frac-tional Calculus and Applied Analysis vol 16 no 3 pp 613ndash6292013

[30] J Cermak T Kisela and L Nechvatal ldquoStability and asymptoticproperties of a linear fractional difference equationrdquo Advancesin Difference Equations vol 2012 article 122 2012

[31] J Cermak T Kisela and L Nechvatal ldquoStability regions forlinear fractional differential systems and their discretizationsrdquoApplied Mathematics and Computation vol 219 no 12 pp7012ndash7022 2013

[32] D Mozyrska and E Girejko ldquoOverview of the fractional h-difference operatorsrdquo in Advances in Harmonic Analysis andOperator Theory vol 229 pp 253ndash267 Birkhauser 2013

[33] T Abdeljawad ldquoOn Riemann and Caputo fractional differ-encesrdquo Computers ampMathematics with Applications vol 62 no3 pp 1602ndash1611 2011

[34] D Mozyrska and E Pawluszewicz ldquoControllability of h-dif-ference linear control systems with two fractional ordersrdquoInternational Journal of Systems Science vol 46 no 4 pp 662ndash669 2015

[35] E Pawluszewicz and D Mozyrska ldquoConstrained controllabilityof h-difference linear systems with two fractional ordersrdquo inAdvances in the Theory and Applications of Non-Integer OrderSystems W Mitkowski J Kacprzyk and J Baranowski Edsvol 257 of Lecture Notes in Electrical Engineering pp 67ndash75Springer Berlin Germany 2013

[36] T Kaczorek ldquoReachability and controllability to zero tests forstandard and positive fractional discrete-time systemsrdquo JournalEuropeen des SystemesAutomatises vol 42 no 6ndash8 pp 769ndash7872008

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Table 1 BasicZ-transform formulas used in the text

119910 = 119910 (119899) Z[119910](119911)

120593120572(119899) = (

119899 + 120572 minus 1

119899) = (minus1)

119899

(minus120572

119899) (

119911

119911 minus 1)120572

(119886Δminus120572

ℎ119909)(119886 + 120572ℎ + 119899ℎ) (

ℎ119911

119911 minus 1)

120572

119883(119911)

120593lowast119896120572

(119899)1

119911119896(

119911

119911 minus 1)119896120572+1

(119886Δ120572

ℎlowast119909)(119886 + (1 minus 120572)ℎ + 119899ℎ) 119911 (

ℎ119911

119911 minus 1)

minus120572

119883(119911) minus 119911ℎminus120572

(119911

119911 minus 1)1minus120572

119909(119886)

119864(120572120573)

(120582 119899) =

infin

sum119896=0

120582119896

120593119896120572+120573

(119899 minus 119896) (119911

119911 minus 1)120573

(1 minus120582

119911(

119911

119911 minus 1)120572

)

minus1

120593119896120572

(119899)1

119911119896(

119911

119911 minus 1)(119896+1)120572

(119886Δ120572

ℎ119909)(119886 + (1 minus 120572)ℎ + 119899ℎ) 119911 (

ℎ119911

119911 minus 1)

minus120572

119883(119911) minus 119911ℎminus120572

119909(119886)

(0Δ120572

ℎ119910)(119899ℎ + ℎ) 119911 (

ℎ119911

119911 minus 1)

minus120572

119883(119911) minus 119911ℎminus120572

119909(119886)

There is the possible use of the generalized Laplacetransformation on time scales for that see for example[26] Such method was used in [16] and [27] under themethod called the 119877-transform and the Laplace transformrespectively for the situation where on the right side of thesystem there are functions that depend only on time Addi-tionally in the dissertation [28] the possible use of the Laplacetransform for semilinear systems ismentionedMore classicalsolution of the problem is also presented in [29] wherethe authors use the Z-transform to give the conditions forstability of systems with the Caputo-type operator Theconclusions were derived based on the image in the complexdomainThemain advantage of the use of theZ-transform isto introduce the natural language for discrete systems itmeans to work with sequences instead of discrete functionsdefined on various domains

The stability property is the main issue in dynamicalsystems As themost of descriptions of behaviour of processesare digital due to computing tools used at computers it isimportant to know some conditions for stability of discrete-time systemsThe results are similar for different operators In[30 31] the study of the stability problem for discrete frac-tional systems with the nabla Riemann-Liouville-type differ-ence operator is presented The authors stress that the Z-transform can be used as a very effective method for stabilityanalysis of systems In engineering problems more oftensystems are usedwith theGrunwald-Letnikov-type fractionaldifference One of the attempts to the stability of fractionaldifference systems is the notion of practical stability see [6 7]Since the linear systems are defined by some matrices theconditions connected with eigenvalues of these matrices arepresented for instance in [11 13 14]

Our paper is rather motivated by the idea of presentingimages via Z-transform of fractional difference operatorsand consequences into stability property We receive resultsfor stability of linear systems with three operators and with

steps ℎ gt 0 In the case of the Grunwald-Letnikov-type frac-tional operator our results are similar to those in [11 13 14]However steps in the proof of stability condition are based on[31]

The paper is organized as follows In Section 2 we presentthe preliminary material needed for further reading Solu-tions to the considered linear initial value problems with theCaputo-type fractional difference and the image of theunknown vector function are constructed in Section 3 As thesimple consequence of considering two types of solutionone from the classical approach and the second from theZ-transform we can write the images of the one-parameterMittag-Leffler functions Sections 4 and 5 deal with imagesof the unknown vector functions for linear initial valueproblems with the Riemann-Liouville- and the Grunwald-Letnikov-type fractional differences respectively Some com-parisons for theZ-transforms of Caputo-type and Riemann-Liouville-type operators are presented in Section 6 In Sec-tion 7 we study the stability of the considered systemsSection 8 provides the brief conclusions Additionally Table 1presents basicZ-transform formulas used in the text

2 Preliminaries

Now we recall the necessary definitions and technical propo-sitions that are used in the sequel therein the paper Let ℎ gt 0119886 isin R anddefine (ℎN)

119886= 119886 119886+ℎ 119886+2ℎ and120590(119905) = 119905+ℎ

for any 119905 isin (ℎN)119886 Hence 119905 = 119886 + 119899ℎ for 119899 isin N

0 For a

function 119909 (ℎN)119886

rarr R the forward ℎ-difference operator isdefined as (Δ

ℎ119909)(119905) = (119909(120590(119905))minus119909(119905))ℎ where 119905 isin (ℎN)

119886 see

[21 22] For an arbitrary 120572 isin R and 119905 isin R such that 119905ℎ isin

R minus1 minus2 the ℎ-factorial function is defined by 119905(120572)

ℎ=

ℎ120572(Γ((119905ℎ) + 1)Γ((119905ℎ) + 1 minus 120572)) where Γ is the Euler gammafunction see [21 22] We use the convention that division ata pole yields zero For ℎ = 1 we write shortly 119905

(120572) instead of119905(120572)

1and Δ instead of Δ

1




119886Δminus120572

ℎ119909)(119905) =

(1Γ(120572))sum119899

119904=0(119905 minus 120590(119886 + 119904ℎ))

(120572minus1)

ℎ119909(119886 + 119904ℎ)ℎ where 119905 = 119886 +


119886Δ0

ℎ119909)(119905) = 119909(119905)



119886Δminus120572ℎ119909 (ℎN)


119905 = 119886 + (120572 + 119899)ℎ

(119886Δminus120572

ℎ119909) (119905) = ℎ

120572

119899

sum119904=0

Γ (120572 + 119899 minus 119904)

Γ (120572) Γ (119899 minus 119904 + 1)119909 (119886 + 119904ℎ)

= ℎ120572

119899

sum119904=0

(119899 minus 119904 + 120572 minus 1

119899 minus 119904)119909 (119904)

(1)


120583(119899) = ( 119899+120583minus1

119899) for 119899 isin N

0


120593120572(119899) = (

119899 + 120572 minus 1

119899) = (minus1)

119899

(minus120572

119899) (2)


and 119909 In fact

(119886Δminus120572

ℎ119909) (119905) = ℎ

120572

(120593120572lowast 119909) (119899) (3)


lowast

119909)(119899) = sum119899



119910(119899)119899isinN0


suminfin



120572](119911) = sum

infin

119896=0( 119896+120572minus1119896

) 119911minus119896 =

suminfin

119896=0(minus1)119896



Let 119910(119899)119899isinN0


= suminfin

119904=0|119910(119904)| and

119910infin

= sup119899isinN0





120572


119904=0|120593120572(119904)| = 120593

1205721lt infin



0such that


119904) gt 0

Therefore

100381710038171003817100381712059312057210038171003817100381710038171 =

infin

sum119904=0

1003816100381610038161003816120593120572 (119904)1003816100381610038161003816 =

119896

sum119904=0

(120573

119904) +

infin

sum119904=119896+1

(minus1)119904

(120573

119904)

=

119896

sum119904=0

(120573

119904) +

infin

sum119904=0

(minus1)119904

(120573

119904) minus

119896

sum119904=0

(minus1)119904

(120573

119904)

= 2

[1198962]

sum119904=0

(120573

2119904 + 1) lt infin

(4)


0)


0)


1003816100381610038161003816120593120572 (119899)1003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816


119899

10038161003816100381610038161003816100381610038161003816


119899= 1

(5)



0)


= sup119899isinN0

|120593120572(119899)| lt infin


let one denote119910(119899) = (

119886Δminus120572

ℎ119909)(119905) and 119909(119899) = 119909(119886 + 119899ℎ) Then

Z [119910] (119911) = (ℎ119911

119911 minus 1)

120572

119883 (119911) (6)

where 119883(119911) = Z[119909](119911)


ℎ119909)(119905) =

ℎ120572(120593120572lowast 119909)(119899) then Z[119910](119911) = ℎ120572Z[120593

120572](119911)119883(119911) Moreover


(6)


119909](119911) =







Δminus120572

ℎ(119886Δminus120573

ℎ119909))(119905) = (

119886Δminus(120572+120573)

ℎ119909)(119905) =

(119886+120572ℎ

Δminus120573

ℎ(119886Δminus120572

ℎ119909))(119905) where 119905 isin (ℎN)

119886+(120572+120573)ℎ


Lemma 6 Let 119909 (ℎN)119886


ℎ119909)(119905) = ℎ120572(

119886ℎΔminus120572

1119909)(119905ℎ) where 119905 isin (ℎN)

119886+120572ℎand

119909(119904) = 119909(119904ℎ)


119886Δminus120572

ℎ120595)(119905) = (Γ(120583 + 1)Γ(120583 +

120572+1))(119905minus119886+120583ℎ)(120583+120572)

ℎ where120595(119903) = (119903minus119886+120583ℎ)

(120583)

ℎ 119903 isin (ℎN)

119886

119905 isin (ℎN)119886+120572ℎ


0 we get (

119886Δminus120572

ℎ1)(119905) = (1Γ(120572 +

1))(119905 minus 119886)(120572)

ℎ= (Γ(119899 + 120572 + 1)Γ(120572 + 1)Γ(119899 + 1))ℎ120572 = ( 119899+120572

119899) ℎ120572






E(120572120573)

(120582 119911)

=

infin

sum119896=0

120582119896 (119911 + (119896 minus 1) (120572 minus 1))

(119896120572)

(119911 + 119896 (120572 minus 1))(120573minus1)

Γ (120572119896 + 120573)(7)


119864(120572120573)

(120582 119899) =

infin

sum119896=0

120582119896

120593119896120572+120573

(119899 minus 119896) (8)


(120582 119899) =

suminfin

119896=0120582119896120593119896120572+120572

(119899 minus 119896) and 119864(120572)

(120582 119899) = 119864(1205721)

(120582 119899) =

suminfin

119896=0120582119896120593119896120572+1


(120582 119899) =

119864(1)

(120582 119899) = (1 + 120582)119899



Proposition 8 Let 119864(120572120573)


Z [119864(120572120573)

(120582 sdot)] (119911) = (119911

119911 minus 1)120573

(1 minus120582

119911(

119911

119911 minus 1)120572

)

minus1

(9)



Z [119864(120572120573)

(120582 sdot)] (119911) =

infin

sum119896=0

infin

sum119899=0

120582119896

(119899 minus 119896 + 119896120572 + 120573 minus 1

119899 minus 119896) 119911minus119899

=

infin

sum119896=0

120582119896

119911minus119896

infin

sum119904=0

(119904 + 119896120572 + 120573 minus 1

119904) 119911minus119904

=

infin

sum119896=0

(120582

119911)

119896 infin

sum119904=0

(minus1)119904

(minus119896120572 minus 120573

119904) 119911minus119904

= (119911

119911 minus 1)120573 infin

sum119896=0

(120582

119911)

119896

(119911

119911 minus 1)119896120572

= (119911

119911 minus 1)120573

(1 minus120582

119911(

119911

119911 minus 1)120572

)

minus1

(10)


|120582|


(119911 minus 1)120572

= 120582119911120572minus1

(11)


119864(120572120573)

(120582 119899) = 0




120572 lt 120582 lt 0




119896120572 Z rarr R


values

120593lowast

119896120572(119899) =

(119899 minus 119896 + 119896120572

119899 minus 119896) for 119899 isin N

119896

0 for 119899 lt 119896

(12)



Z [120593lowast

119896120572] (119911) =

1

119911119896(

119911

119911 minus 1)119896120572+1

(13)

for 119911 such that |119911| gt 1


Z [120593lowast

119896120572] (119911)

=

infin

sum119899=0

120593lowast

119896120572(119899) 119911minus119899

=

infin

sum119899=119896

(119899 minus 119896 + 119896120572

119899 minus 119896) 119911minus119899

= 119911minus119896

infin

sum119904=0

(119904 + 119896120572

119904) 119911minus119904

= 119911minus119896

infin

sum119904=0

(minus1)119904

(minus119896120572 minus 1

119904) 119911minus119904

= 119911minus119896

(1 minus1

119911)minus119896120572minus1

=1

119911119896(

119911

119911 minus 1)119896120572+1

(14)



relations hold

(1) 120593lowast0120572

(119899) = 1 120593lowast119896120572

(119896) = 1 and 120593lowast1120572

(1) = 1(2) 120593lowast119896120572



(3) 120593lowast119896120572

(119899) = (1Γ(119896120572 + 1))(119899 minus 119896 + 119896120572)(119896120572) = ( 119899minus119896+119896120572119896120572

)

(4) Z[120593lowast

1120572](119911) = (1119911)(119911(119911 minus 1))

120572+1

= (1(119911 minus 1))(119911(119911 minus

1))120572

(5) Z[120593lowast119896120572

](119911) = (1119911119896) sdot (119911(119911 minus 1))119896120572+1 = (1(119911 minus 1)119896

) sdot



(119899) = ( 119899119896) and Δminus1 ( 119899minus1

119896) =

( 119899119896+1

)



119896)] (119911) =


1199112(119911 minus 1)119896+2 andZ [( 119899

119896+1)] (119911) = (1119911119896+1) sdot (119911(119911 minus 1))119896+2 =


119896)] (119911) = Z [( 119899

119896+1)] (119911) but



)] (119911) =


Z [Δminus1 ( 119899minus1119896

)] (119911) = Z [( 119899119896+1




(119860 119899 + ]) = suminfin

119896=0119860119896

120593lowast

119896120572(119899) = 119864

(120572)(119860 119899) where


(119860 119899) = sum119899

119896=0119860119896120593lowast119896120572


(120572)(119860 119899)


(119860 119899) = (119868 + 119860)119899



lowast





0one has

(Δminus120572

120593lowast

119896120572) (119899 + ]) = 120593

lowast

119896+1120572(119899) (15)



119886

Δ120572


(ℎN)119886


(119886

Δ120572

ℎlowast119909) (119905) = (


ℎ(Δℎ119909)) (119905) (16)


Note that119886

Δ120572




Δ1

ℎlowast119909)(119905) = (

119886Δ0

ℎ(Δℎ119909))(119905) = (Δ

ℎ119909)(119905) where 119905 isin (ℎN)

119886



(119886

Δ120572

ℎlowast119909) (119905) = ℎ

minus120572

(1205931minus120572

lowast Δ119909) (119899) (17)




119886The following


Δ120572


120572ℎ


Lemma 15 (see [21]) Let 119909 (ℎN)119886


Δ120572

ℎlowast119909)(119905) = ℎminus120572(

119886ℎΔ1205721lowast

119909)(119905ℎ) where 119905 isin (ℎN)119886+120572ℎ

and119909(119904) = 119909(119904ℎ)


Δ120572

lowast=

119886ℎ

Δ120572



(Δminus120572

(119886ℎ

Δ120572

lowast119909)) (119899 + 120572) = 119909 (119899 + 120572) minus 119909 (119886) (18)



(119886

Δ120572



Then

Z [119910] (119911) = ℎminus120572

(119911

119911 minus 1)1minus120572

((119911 minus 1)119883 (119911) minus 119911119909 (119886)) (19)

where 119883(119911) = Z[119909](119911) and 119909(119899) = 119909(119886 + 119899ℎ)


Δ120572

ℎlowast119909)(119905) = ℎminus120572(120593

1minus120572lowast







1the


(0Δ120572

lowast120593lowast

119896120572) (119899 + 1 minus 120572) = 120593

lowast

119896minus1120572(119899) (20)



)(119899+1minus120572) =



(119886

Δ120572

ℎlowast119909) (119899ℎ) = 119860119909 (119899ℎ + 119886) 119899 isin N

0 (21)

where 119909 (ℎZ)119886




0can be


( ]Δ120572

lowast119909) (119899) = 119860119909 (119899 + ]) 119899 isin N

0 (22a)

119909 (]) = 1199090isin R119901

(22b)




(119886

Δ120572

ℎlowast119909) (119905) = 119860119909 (119905 + 119886) 119905 isin (ℎN)

0(23a)

119909 (119886) = 1199090 1199090isin R119901 (23b)


119909 (119905) = E(120572)

(119860ℎ120572

119905

ℎ) 1199090= 119864(120572)

(119860ℎ120572

119899) 1199090

=

infin

sum119896=0

119860119896

ℎ119896120572

120593lowast

119896120572(119899) 1199090

(24)


120572minus1



(119860ℎ120572

119899) =

suminfin

119896=0119860119896ℎ119896120572120593lowast




Z [119872] (119911) =119911

119911 minus 1(119868 minus

1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

(25)


1))120572119860ℎ120572)minus1119909






Δ120572

ℎlowast119909)(119905) = 120582ℎminus120572119909(119905 + 119886) 119909(119886) = 1

Then119864(120572)


Zminus1[suminfin



120572 = 120582 into (25)




lowast





120593119896120572

(119899) =

(119899 minus 119896 + 119896120572 + 120572 minus 1

119899 minus 119896) for 119899 isin N

119896

0 for 119899 lt 119896

(26)

Proposition 19 Let 120593119896120572


Z [120593119896120572

] (119911) =1

119911119896(

119911

119911 minus 1)119896120572+120572

(27)

for 119911 such that |119911| gt 1


Z [120593119896120572

] (119911) =

infin

sum119899=0

120593119896120572

(119899) 119911minus119899

=

infin

sum119899=119896

(119899 minus 119896 + (119896 + 1) 120572 minus 1

119899 minus 119896) 119911minus119899

= 119911minus119896

infin

sum119904=0

(119904 + (119896 + 1) 120572 minus 1

119904) 119911minus119904

= 119911minus119896

infin

sum119904=0

(minus1)119904

(minus (119896 + 1) 120572

119904) 119911minus119904

= 119911minus119896

(1 minus1

119911)minus119896120572minus120572

=1

119911119896(

119911

119911 minus 1)(119896+1)120572

(28)



relations hold

(1) 1205930120572

(0) = 1 120593119896120572

(119896) = 1 and 1205931120572

(1) = 1(2) 120593119896120572

(119899) ge 0 and 120593lowast119896120572

(119899) ge 0

(3) 120593119896120572

(119899) = (119899 minus (119896 + 1) + (119896 + 1)120572)((119896+1)120572minus1)Γ((119896 + 1)120572)(4) Z[120593

0120572](119911) = (119911(119911 minus 1))120572Z[120593

1120572](119911) = (1119911)(119911(119911 minus

1))2120572(5) Z[120593

119896120572](119911) = (1119911119896)(119911(119911 minus 1))(119896+1)120572 for 119896 isin N

0

(6) For 120572 = 1 we have 1205931198961

(119899) = 120593lowast1198961

(119899) = ( 119899119896)




(119860 119899+]) = suminfin

119896=0119860119896120593119896120572

(119899) = 119864(120572120572)

(119860 119899) where ] = 120572minus

1 Note that in fact 119864(120572120572)

(119860 119899) = sum119899

119896=0119860119896120593119896120572


(120572120572)(119860 119899) always



(119860 119899) = (119868 + 119860)119899






one has

(Δminus120572

120593119896120572

) (119899 + ]) = 120593119896+1120572

(119899) (29)


)(119899 +

])](119911) = (1119911)(119911(119911 minus 1))120572 sdot Z[120593119896120572



119896120572)(119899+

])](119911) = (1119911119896+1)(119911(119911 minus 1))(119896+2)120572 = Z[120593119896+1120572




119886Δ120572


119909 (ℎN)119886


(119886Δ120572

ℎ119909) (119905) = (Δ


ℎ119909)) (119905) (30)



(119886

Δ120572

ℎlowast119909) (119905) = (

119886Δ120572

ℎ119909) (119905) minus

119909 (119886) sdot (119905 minus 119886)(minus120572)

ℎ

Γ (1 minus 120572)

= (119886Δ120572

ℎ119909) (119905) minus

119909 (119886)

ℎ120572(

119905 minus 119886

ℎminus120572

)

(31)


119886

Δ1

ℎlowast119909)(119905) =

(119886Δ1

ℎ119909)(119905) = (Δ

ℎ119909)(119905)





ℎ(119886Δ120572ℎ119909))(119905) = 119909(119905) minus 119909(119886) sdot

(ℎ1minus120572Γ(120572))119905(120572minus1)

ℎ= 119909(119905) minus 119909(119886) sdot ( 119905ℎ

120572minus1) 119905 isin (ℎN)

120572ℎ


Lemma 23 Let 119909 (ℎN)119886


ℎ119909)(119905) = ℎminus120572(

119886ℎΔ120572

1119909)(119905ℎ) where 119905 isin (ℎN)

119886+(1minus120572)ℎand

119909(119904) = 119909(119904ℎ)


Δ120572

=119886ℎ

Δ120572

1 Then the


(Δminus120572

(119886ℎ

Δ120572

119909)) (119899 + 120572) = 119909 (119899 + 120572) minus 119909 (119886) sdot (119899 + 120572

119899 + 1)

(32)



(119886Δ120572

ℎ119909)(119905) where 119905 isin (ℎN)


Then

Z [119910] (119911) = 119911 (ℎ119911

119911 minus 1)

minus120572

119883(119911) minus 119911ℎminus120572

119909 (119886) (33)

where 119883(119911) = Z[119909](119911) and 119909(119899) = 119909(119886 + 119899ℎ)


minus(1minus120572)

119909)(119899) Then Z[119910](119911) =




ℎof 119909



(0Δ120572

120593119896120572

) (119899 + 1 minus 120572) = 120593119896minus1120572

(119899) (34)

while for 119896 = 0 we get 1205930120572

(119899) = ( 119899+120572minus1119899

) = 120593120572(119899) and


1205930120572

)(119899+1minus120572) = (0Δ120572

120593120572)(119899+1minus120572) = 0


(119886Δ120572

ℎ119909) (119899ℎ) = 119860119909 (119899ℎ + 119886) 119899 isin N

0 (35)

where 119909 (ℎZ)119886


0isin R119901


( ]Δ120572

119909) (119899) = 119860119909 (119899 + ]) 119899 isin N0 (36a)

119909 (]) = 1199090isin R119901

(36b)



(119886Δ120572

ℎ119909) (119905) = 119860119909 (119905 + 119886) 119905 isin (ℎN)

0 (37a)

119909 (119886) = 1199090 1199090isin R119901

(37b)



119909 (119905) = E(120572120572)

(119860ℎ120572

119905

ℎ) 1199090= 119864(120572120572)

(119860ℎ120572

119899) 1199090

=

infin

sum119896=0

119860119896

ℎ119896120572

120593119896120572

(119899) 1199090

(38)


120572minus1




Z [119872] (119911) = (119911

119911 minus 1)120572

(119868 minus1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

(39)


minus120572





Let 119909 (ℎN)119886


(119886Δ120572






119864(120572120572)


119896=0((120582119896119911119896)(119911(119911 minus 1))119896120572+120572)](119899)





119886Δ120572


119886rarr

R is defined by

(119886Δ120572

ℎ119909) (119905) =

(119905minus119886)ℎ

sum119904=0

119886(120572)

119904119909 (119905 minus 119904ℎ) (40)

where

119886(120572)

119904= (minus1)

119904

(120572

119904)(

1

ℎ120572) with

(120572

119904) =

1 for 119904 = 0


119904 for 119904 isin N

(41)


(119886Δ120572

ℎ119909)(119905) where 119905 isin (ℎN)

119886and 119905 = 119886 + 119899ℎ 119899 isin N

0 Then

Z [119910] (119911) = (ℎ119911

119911 minus 1)

minus120572

119883 (119911) (42)

where 119883(119911) = Z[119909](119911) and 119909(119899) = 119909(119886 + 119899ℎ)


ℎ119909](119911) = Z[119886(120572)

119904lowast119909][119911] =

Z[119886(120572)119904




ℎ119909)(119899ℎ) =

(0Δ120572


0 or 119909(119905) =




ℎ119910)(119905 + ℎ) =

(119886Δ120572


119886

Proof Let119891(119899) = (0Δ120572



ℎ119910)(0) = 119911(ℎ119911(119911minus


Z[119892] where 119883(119911) = Z[119909](119911) 119909(119899) = 119910(119899ℎ) = 119909(119886 + 119899ℎ) and119892(119899) = (

119886Δ120572

ℎ119909)(119905) 119905 isin (ℎN)




0Δ120572

ℎ119910)(119899ℎ + ℎ) as for the




(0Δ120572

119910) (119905 + ℎ) = 119860119910 (119905) 119905 isin (ℎN)0

119910 (0) = 1199100 1199100isin R119901

(43)




(119860ℎ120572 + 119868120572)Φ(119905) with Φ(0) = 119868



(0Δ120572

ℎ119910) (119905 + ℎ) = 119860119910 (119905) 119905 isin (ℎN)

0

119910 (0) = 1199100 1199100isin R119901

(44)


119910 (119905) = 119864(120572120572)

(119860ℎ120572

119905

ℎ) 1199100 (45)




Φ (119899ℎ) = Zminus1

[(119911

119911 minus 1)120572

(119868 minus1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

] (119899)

(46)




ℎminus119886

Δ120572

ℎlowast)119909)(119905) where 119905 =

119886 + (1 minus 120572)ℎ + 119899ℎ Then

Z [119910] (119911) = ℎminus120572

119911((119911

119911 minus 1)1minus120572

minus 1)119909 (119886) (47)



)] (119911)119909(119886) =


119904=0(minus1)119904

( 120572minus1119904


1)119909(119886)



119864(120572120572)

(119860 119899) = (119864(120572)


) (119899) (48)


Z [119864(120572)


] (119911)

= Z [119864(120572)

(119860 sdot)] (119911)Z [120593120572minus1

] (119911)

=119911

119911 minus 1(119868 minus

1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

(119911

119911 minus 1)120572minus1

= (119911

119911 minus 1)120572

(119868 minus1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

(49)



](119911) =

Z[119864(120572120572)





(119886Δ120572

lowast119909) (119899) = 119860119909 (119886 + 119899) + 119861119906 (119899)

119910 (119899) = 119862119909 (119886 + 119899) + 119863119906 (119899) (50)


(119886Δ120572

119909) (119899) = 119860119909 (119886 + 119899) + 119861119906 (119899)

119910 (119899) = 119862119909 (119886 + 119899) + 119863119906 (119899)(51)




1minus120572

(119911minus

1)119883(119911) = 119860119883(119911) + 119861119880(119911) Then 119883(119911) = (1(119911 minus 1)) sdot



120572119862(119868minus


(120572120572)(119860 119899minus1)119861We


(minus1) = 0




lowast119909eq)(119905) = 119891(119905 119909eq) = 0 (and (

119886Δ120572119894 119909eq)(119905) = 119891(119905 119909eq)


0




0


implies

lim119896rarrinfin

119909 (119886 + 119896ℎ) = 0 (52)




lowast



02

01

0

minus01

minus02

50 100 150 200xvalues

n steps


02

01

0

minus01

minus02

50 100 150 200yvalues

n steps


y

x

02

02

01

01

0

0

minus01

minus01

minus02

minus02




119909(119886 + 119899ℎ) =

lim119899rarrinfin

1205930120572




det(119868 minus1

119911(

119911

119911 minus 1)120572

119860ℎ120572

) = 0 (53)







(1119911)(119911(119911 minus 1))120572

119869ℎ120572

)minus1






119904)


119897 119897 = 1 119904 Let 119896

119894isin N1




119894is 119901119894


(

(

119908minus1 (120582119894) 119908minus2 (120582

119894) 119908minus119903119902 (120582

119894)

0 119908minus1 (120582119894) 119908minus119903119902+1 (120582

119894)

d

0 0 119908minus1 (120582119894)

)

)

(54)


det(119908(119860)) = det(119908(119869)) = prod119898

119894=1119908119896119894(120582

119894) and 119883(119911) is formed by




119899rarrinfin119909(119899) = lim

119911rarr1(119911 minus 1)119883(119911) = 0



are not stable



120572 for all120582 isin Spec(119860)


120572minus2120572

2120572minus2120572 ]


119890= 0 Since




1= minus2120572 minus 2120572119894

1205822







0

y

x

times107

times107

15

10

10

05

minus10

minus10

minus05


8 Conclusions




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119886Δminus120572

ℎ119909)(119905) =

(1Γ(120572))sum119899

119904=0(119905 minus 120590(119886 + 119904ℎ))

(120572minus1)

ℎ119909(119886 + 119904ℎ)ℎ where 119905 = 119886 +


119886Δ0

ℎ119909)(119905) = 119909(119905)



119886Δminus120572ℎ119909 (ℎN)


119905 = 119886 + (120572 + 119899)ℎ

(119886Δminus120572

ℎ119909) (119905) = ℎ

120572

119899

sum119904=0

Γ (120572 + 119899 minus 119904)

Γ (120572) Γ (119899 minus 119904 + 1)119909 (119886 + 119904ℎ)

= ℎ120572

119899

sum119904=0

(119899 minus 119904 + 120572 minus 1

119899 minus 119904)119909 (119904)

(1)


120583(119899) = ( 119899+120583minus1

119899) for 119899 isin N

0


120593120572(119899) = (

119899 + 120572 minus 1

119899) = (minus1)

119899

(minus120572

119899) (2)


and 119909 In fact

(119886Δminus120572

ℎ119909) (119905) = ℎ

120572

(120593120572lowast 119909) (119899) (3)


lowast

119909)(119899) = sum119899



119910(119899)119899isinN0


suminfin



120572](119911) = sum

infin

119896=0( 119896+120572minus1119896

) 119911minus119896 =

suminfin

119896=0(minus1)119896



Let 119910(119899)119899isinN0


= suminfin

119904=0|119910(119904)| and

119910infin

= sup119899isinN0





120572


119904=0|120593120572(119904)| = 120593

1205721lt infin



0such that


119904) gt 0

Therefore

100381710038171003817100381712059312057210038171003817100381710038171 =

infin

sum119904=0

1003816100381610038161003816120593120572 (119904)1003816100381610038161003816 =

119896

sum119904=0

(120573

119904) +

infin

sum119904=119896+1

(minus1)119904

(120573

119904)

=

119896

sum119904=0

(120573

119904) +

infin

sum119904=0

(minus1)119904

(120573

119904) minus

119896

sum119904=0

(minus1)119904

(120573

119904)

= 2

[1198962]

sum119904=0

(120573

2119904 + 1) lt infin

(4)


0)


0)


1003816100381610038161003816120593120572 (119899)1003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816


119899

10038161003816100381610038161003816100381610038161003816


119899= 1

(5)



0)


= sup119899isinN0

|120593120572(119899)| lt infin


let one denote119910(119899) = (

119886Δminus120572

ℎ119909)(119905) and 119909(119899) = 119909(119886 + 119899ℎ) Then

Z [119910] (119911) = (ℎ119911

119911 minus 1)

120572

119883 (119911) (6)

where 119883(119911) = Z[119909](119911)


ℎ119909)(119905) =

ℎ120572(120593120572lowast 119909)(119899) then Z[119910](119911) = ℎ120572Z[120593

120572](119911)119883(119911) Moreover


(6)


119909](119911) =







Δminus120572

ℎ(119886Δminus120573

ℎ119909))(119905) = (

119886Δminus(120572+120573)

ℎ119909)(119905) =

(119886+120572ℎ

Δminus120573

ℎ(119886Δminus120572

ℎ119909))(119905) where 119905 isin (ℎN)

119886+(120572+120573)ℎ


Lemma 6 Let 119909 (ℎN)119886


ℎ119909)(119905) = ℎ120572(

119886ℎΔminus120572

1119909)(119905ℎ) where 119905 isin (ℎN)

119886+120572ℎand

119909(119904) = 119909(119904ℎ)


119886Δminus120572

ℎ120595)(119905) = (Γ(120583 + 1)Γ(120583 +

120572+1))(119905minus119886+120583ℎ)(120583+120572)

ℎ where120595(119903) = (119903minus119886+120583ℎ)

(120583)

ℎ 119903 isin (ℎN)

119886

119905 isin (ℎN)119886+120572ℎ


0 we get (

119886Δminus120572

ℎ1)(119905) = (1Γ(120572 +

1))(119905 minus 119886)(120572)

ℎ= (Γ(119899 + 120572 + 1)Γ(120572 + 1)Γ(119899 + 1))ℎ120572 = ( 119899+120572

119899) ℎ120572






E(120572120573)

(120582 119911)

=

infin

sum119896=0

120582119896 (119911 + (119896 minus 1) (120572 minus 1))

(119896120572)

(119911 + 119896 (120572 minus 1))(120573minus1)

Γ (120572119896 + 120573)(7)


119864(120572120573)

(120582 119899) =

infin

sum119896=0

120582119896

120593119896120572+120573

(119899 minus 119896) (8)


(120582 119899) =

suminfin

119896=0120582119896120593119896120572+120572

(119899 minus 119896) and 119864(120572)

(120582 119899) = 119864(1205721)

(120582 119899) =

suminfin

119896=0120582119896120593119896120572+1


(120582 119899) =

119864(1)

(120582 119899) = (1 + 120582)119899



Proposition 8 Let 119864(120572120573)


Z [119864(120572120573)

(120582 sdot)] (119911) = (119911

119911 minus 1)120573

(1 minus120582

119911(

119911

119911 minus 1)120572

)

minus1

(9)



Z [119864(120572120573)

(120582 sdot)] (119911) =

infin

sum119896=0

infin

sum119899=0

120582119896

(119899 minus 119896 + 119896120572 + 120573 minus 1

119899 minus 119896) 119911minus119899

=

infin

sum119896=0

120582119896

119911minus119896

infin

sum119904=0

(119904 + 119896120572 + 120573 minus 1

119904) 119911minus119904

=

infin

sum119896=0

(120582

119911)

119896 infin

sum119904=0

(minus1)119904

(minus119896120572 minus 120573

119904) 119911minus119904

= (119911

119911 minus 1)120573 infin

sum119896=0

(120582

119911)

119896

(119911

119911 minus 1)119896120572

= (119911

119911 minus 1)120573

(1 minus120582

119911(

119911

119911 minus 1)120572

)

minus1

(10)


|120582|


(119911 minus 1)120572

= 120582119911120572minus1

(11)


119864(120572120573)

(120582 119899) = 0




120572 lt 120582 lt 0




119896120572 Z rarr R


values

120593lowast

119896120572(119899) =

(119899 minus 119896 + 119896120572

119899 minus 119896) for 119899 isin N

119896

0 for 119899 lt 119896

(12)



Z [120593lowast

119896120572] (119911) =

1

119911119896(

119911

119911 minus 1)119896120572+1

(13)

for 119911 such that |119911| gt 1


Z [120593lowast

119896120572] (119911)

=

infin

sum119899=0

120593lowast

119896120572(119899) 119911minus119899

=

infin

sum119899=119896

(119899 minus 119896 + 119896120572

119899 minus 119896) 119911minus119899

= 119911minus119896

infin

sum119904=0

(119904 + 119896120572

119904) 119911minus119904

= 119911minus119896

infin

sum119904=0

(minus1)119904

(minus119896120572 minus 1

119904) 119911minus119904

= 119911minus119896

(1 minus1

119911)minus119896120572minus1

=1

119911119896(

119911

119911 minus 1)119896120572+1

(14)



relations hold

(1) 120593lowast0120572

(119899) = 1 120593lowast119896120572

(119896) = 1 and 120593lowast1120572

(1) = 1(2) 120593lowast119896120572



(3) 120593lowast119896120572

(119899) = (1Γ(119896120572 + 1))(119899 minus 119896 + 119896120572)(119896120572) = ( 119899minus119896+119896120572119896120572

)

(4) Z[120593lowast

1120572](119911) = (1119911)(119911(119911 minus 1))

120572+1

= (1(119911 minus 1))(119911(119911 minus

1))120572

(5) Z[120593lowast119896120572

](119911) = (1119911119896) sdot (119911(119911 minus 1))119896120572+1 = (1(119911 minus 1)119896

) sdot



(119899) = ( 119899119896) and Δminus1 ( 119899minus1

119896) =

( 119899119896+1

)



119896)] (119911) =


1199112(119911 minus 1)119896+2 andZ [( 119899

119896+1)] (119911) = (1119911119896+1) sdot (119911(119911 minus 1))119896+2 =


119896)] (119911) = Z [( 119899

119896+1)] (119911) but



)] (119911) =


Z [Δminus1 ( 119899minus1119896

)] (119911) = Z [( 119899119896+1




(119860 119899 + ]) = suminfin

119896=0119860119896

120593lowast

119896120572(119899) = 119864

(120572)(119860 119899) where


(119860 119899) = sum119899

119896=0119860119896120593lowast119896120572


(120572)(119860 119899)


(119860 119899) = (119868 + 119860)119899



lowast





0one has

(Δminus120572

120593lowast

119896120572) (119899 + ]) = 120593

lowast

119896+1120572(119899) (15)



119886

Δ120572


(ℎN)119886


(119886

Δ120572

ℎlowast119909) (119905) = (


ℎ(Δℎ119909)) (119905) (16)


Note that119886

Δ120572




Δ1

ℎlowast119909)(119905) = (

119886Δ0

ℎ(Δℎ119909))(119905) = (Δ

ℎ119909)(119905) where 119905 isin (ℎN)

119886



(119886

Δ120572

ℎlowast119909) (119905) = ℎ

minus120572

(1205931minus120572

lowast Δ119909) (119899) (17)




119886The following


Δ120572


120572ℎ


Lemma 15 (see [21]) Let 119909 (ℎN)119886


Δ120572

ℎlowast119909)(119905) = ℎminus120572(

119886ℎΔ1205721lowast

119909)(119905ℎ) where 119905 isin (ℎN)119886+120572ℎ

and119909(119904) = 119909(119904ℎ)


Δ120572

lowast=

119886ℎ

Δ120572



(Δminus120572

(119886ℎ

Δ120572

lowast119909)) (119899 + 120572) = 119909 (119899 + 120572) minus 119909 (119886) (18)



(119886

Δ120572



Then

Z [119910] (119911) = ℎminus120572

(119911

119911 minus 1)1minus120572

((119911 minus 1)119883 (119911) minus 119911119909 (119886)) (19)

where 119883(119911) = Z[119909](119911) and 119909(119899) = 119909(119886 + 119899ℎ)


Δ120572

ℎlowast119909)(119905) = ℎminus120572(120593

1minus120572lowast







1the


(0Δ120572

lowast120593lowast

119896120572) (119899 + 1 minus 120572) = 120593

lowast

119896minus1120572(119899) (20)



)(119899+1minus120572) =



(119886

Δ120572

ℎlowast119909) (119899ℎ) = 119860119909 (119899ℎ + 119886) 119899 isin N

0 (21)

where 119909 (ℎZ)119886




0can be


( ]Δ120572

lowast119909) (119899) = 119860119909 (119899 + ]) 119899 isin N

0 (22a)

119909 (]) = 1199090isin R119901

(22b)




(119886

Δ120572

ℎlowast119909) (119905) = 119860119909 (119905 + 119886) 119905 isin (ℎN)

0(23a)

119909 (119886) = 1199090 1199090isin R119901 (23b)


119909 (119905) = E(120572)

(119860ℎ120572

119905

ℎ) 1199090= 119864(120572)

(119860ℎ120572

119899) 1199090

=

infin

sum119896=0

119860119896

ℎ119896120572

120593lowast

119896120572(119899) 1199090

(24)


120572minus1



(119860ℎ120572

119899) =

suminfin

119896=0119860119896ℎ119896120572120593lowast




Z [119872] (119911) =119911

119911 minus 1(119868 minus

1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

(25)


1))120572119860ℎ120572)minus1119909






Δ120572

ℎlowast119909)(119905) = 120582ℎminus120572119909(119905 + 119886) 119909(119886) = 1

Then119864(120572)


Zminus1[suminfin



120572 = 120582 into (25)




lowast





120593119896120572

(119899) =

(119899 minus 119896 + 119896120572 + 120572 minus 1

119899 minus 119896) for 119899 isin N

119896

0 for 119899 lt 119896

(26)

Proposition 19 Let 120593119896120572


Z [120593119896120572

] (119911) =1

119911119896(

119911

119911 minus 1)119896120572+120572

(27)

for 119911 such that |119911| gt 1


Z [120593119896120572

] (119911) =

infin

sum119899=0

120593119896120572

(119899) 119911minus119899

=

infin

sum119899=119896

(119899 minus 119896 + (119896 + 1) 120572 minus 1

119899 minus 119896) 119911minus119899

= 119911minus119896

infin

sum119904=0

(119904 + (119896 + 1) 120572 minus 1

119904) 119911minus119904

= 119911minus119896

infin

sum119904=0

(minus1)119904

(minus (119896 + 1) 120572

119904) 119911minus119904

= 119911minus119896

(1 minus1

119911)minus119896120572minus120572

=1

119911119896(

119911

119911 minus 1)(119896+1)120572

(28)



relations hold

(1) 1205930120572

(0) = 1 120593119896120572

(119896) = 1 and 1205931120572

(1) = 1(2) 120593119896120572

(119899) ge 0 and 120593lowast119896120572

(119899) ge 0

(3) 120593119896120572

(119899) = (119899 minus (119896 + 1) + (119896 + 1)120572)((119896+1)120572minus1)Γ((119896 + 1)120572)(4) Z[120593

0120572](119911) = (119911(119911 minus 1))120572Z[120593

1120572](119911) = (1119911)(119911(119911 minus

1))2120572(5) Z[120593

119896120572](119911) = (1119911119896)(119911(119911 minus 1))(119896+1)120572 for 119896 isin N

0

(6) For 120572 = 1 we have 1205931198961

(119899) = 120593lowast1198961

(119899) = ( 119899119896)




(119860 119899+]) = suminfin

119896=0119860119896120593119896120572

(119899) = 119864(120572120572)

(119860 119899) where ] = 120572minus

1 Note that in fact 119864(120572120572)

(119860 119899) = sum119899

119896=0119860119896120593119896120572


(120572120572)(119860 119899) always



(119860 119899) = (119868 + 119860)119899






one has

(Δminus120572

120593119896120572

) (119899 + ]) = 120593119896+1120572

(119899) (29)


)(119899 +

])](119911) = (1119911)(119911(119911 minus 1))120572 sdot Z[120593119896120572



119896120572)(119899+

])](119911) = (1119911119896+1)(119911(119911 minus 1))(119896+2)120572 = Z[120593119896+1120572




119886Δ120572


119909 (ℎN)119886


(119886Δ120572

ℎ119909) (119905) = (Δ


ℎ119909)) (119905) (30)



(119886

Δ120572

ℎlowast119909) (119905) = (

119886Δ120572

ℎ119909) (119905) minus

119909 (119886) sdot (119905 minus 119886)(minus120572)

ℎ

Γ (1 minus 120572)

= (119886Δ120572

ℎ119909) (119905) minus

119909 (119886)

ℎ120572(

119905 minus 119886

ℎminus120572

)

(31)


119886

Δ1

ℎlowast119909)(119905) =

(119886Δ1

ℎ119909)(119905) = (Δ

ℎ119909)(119905)





ℎ(119886Δ120572ℎ119909))(119905) = 119909(119905) minus 119909(119886) sdot

(ℎ1minus120572Γ(120572))119905(120572minus1)

ℎ= 119909(119905) minus 119909(119886) sdot ( 119905ℎ

120572minus1) 119905 isin (ℎN)

120572ℎ


Lemma 23 Let 119909 (ℎN)119886


ℎ119909)(119905) = ℎminus120572(

119886ℎΔ120572

1119909)(119905ℎ) where 119905 isin (ℎN)

119886+(1minus120572)ℎand

119909(119904) = 119909(119904ℎ)


Δ120572

=119886ℎ

Δ120572

1 Then the


(Δminus120572

(119886ℎ

Δ120572

119909)) (119899 + 120572) = 119909 (119899 + 120572) minus 119909 (119886) sdot (119899 + 120572

119899 + 1)

(32)



(119886Δ120572

ℎ119909)(119905) where 119905 isin (ℎN)


Then

Z [119910] (119911) = 119911 (ℎ119911

119911 minus 1)

minus120572

119883(119911) minus 119911ℎminus120572

119909 (119886) (33)

where 119883(119911) = Z[119909](119911) and 119909(119899) = 119909(119886 + 119899ℎ)


minus(1minus120572)

119909)(119899) Then Z[119910](119911) =




ℎof 119909



(0Δ120572

120593119896120572

) (119899 + 1 minus 120572) = 120593119896minus1120572

(119899) (34)

while for 119896 = 0 we get 1205930120572

(119899) = ( 119899+120572minus1119899

) = 120593120572(119899) and


1205930120572

)(119899+1minus120572) = (0Δ120572

120593120572)(119899+1minus120572) = 0


(119886Δ120572

ℎ119909) (119899ℎ) = 119860119909 (119899ℎ + 119886) 119899 isin N

0 (35)

where 119909 (ℎZ)119886


0isin R119901


( ]Δ120572

119909) (119899) = 119860119909 (119899 + ]) 119899 isin N0 (36a)

119909 (]) = 1199090isin R119901

(36b)



(119886Δ120572

ℎ119909) (119905) = 119860119909 (119905 + 119886) 119905 isin (ℎN)

0 (37a)

119909 (119886) = 1199090 1199090isin R119901

(37b)



119909 (119905) = E(120572120572)

(119860ℎ120572

119905

ℎ) 1199090= 119864(120572120572)

(119860ℎ120572

119899) 1199090

=

infin

sum119896=0

119860119896

ℎ119896120572

120593119896120572

(119899) 1199090

(38)


120572minus1




Z [119872] (119911) = (119911

119911 minus 1)120572

(119868 minus1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

(39)


minus120572





Let 119909 (ℎN)119886


(119886Δ120572






119864(120572120572)


119896=0((120582119896119911119896)(119911(119911 minus 1))119896120572+120572)](119899)





119886Δ120572


119886rarr

R is defined by

(119886Δ120572

ℎ119909) (119905) =

(119905minus119886)ℎ

sum119904=0

119886(120572)

119904119909 (119905 minus 119904ℎ) (40)

where

119886(120572)

119904= (minus1)

119904

(120572

119904)(

1

ℎ120572) with

(120572

119904) =

1 for 119904 = 0


119904 for 119904 isin N

(41)


(119886Δ120572

ℎ119909)(119905) where 119905 isin (ℎN)

119886and 119905 = 119886 + 119899ℎ 119899 isin N

0 Then

Z [119910] (119911) = (ℎ119911

119911 minus 1)

minus120572

119883 (119911) (42)

where 119883(119911) = Z[119909](119911) and 119909(119899) = 119909(119886 + 119899ℎ)


ℎ119909](119911) = Z[119886(120572)

119904lowast119909][119911] =

Z[119886(120572)119904




ℎ119909)(119899ℎ) =

(0Δ120572


0 or 119909(119905) =




ℎ119910)(119905 + ℎ) =

(119886Δ120572


119886

Proof Let119891(119899) = (0Δ120572



ℎ119910)(0) = 119911(ℎ119911(119911minus


Z[119892] where 119883(119911) = Z[119909](119911) 119909(119899) = 119910(119899ℎ) = 119909(119886 + 119899ℎ) and119892(119899) = (

119886Δ120572

ℎ119909)(119905) 119905 isin (ℎN)




0Δ120572

ℎ119910)(119899ℎ + ℎ) as for the




(0Δ120572

119910) (119905 + ℎ) = 119860119910 (119905) 119905 isin (ℎN)0

119910 (0) = 1199100 1199100isin R119901

(43)




(119860ℎ120572 + 119868120572)Φ(119905) with Φ(0) = 119868



(0Δ120572

ℎ119910) (119905 + ℎ) = 119860119910 (119905) 119905 isin (ℎN)

0

119910 (0) = 1199100 1199100isin R119901

(44)


119910 (119905) = 119864(120572120572)

(119860ℎ120572

119905

ℎ) 1199100 (45)




Φ (119899ℎ) = Zminus1

[(119911

119911 minus 1)120572

(119868 minus1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

] (119899)

(46)




ℎminus119886

Δ120572

ℎlowast)119909)(119905) where 119905 =

119886 + (1 minus 120572)ℎ + 119899ℎ Then

Z [119910] (119911) = ℎminus120572

119911((119911

119911 minus 1)1minus120572

minus 1)119909 (119886) (47)



)] (119911)119909(119886) =


119904=0(minus1)119904

( 120572minus1119904


1)119909(119886)



119864(120572120572)

(119860 119899) = (119864(120572)


) (119899) (48)


Z [119864(120572)


] (119911)

= Z [119864(120572)

(119860 sdot)] (119911)Z [120593120572minus1

] (119911)

=119911

119911 minus 1(119868 minus

1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

(119911

119911 minus 1)120572minus1

= (119911

119911 minus 1)120572

(119868 minus1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

(49)



](119911) =

Z[119864(120572120572)





(119886Δ120572

lowast119909) (119899) = 119860119909 (119886 + 119899) + 119861119906 (119899)

119910 (119899) = 119862119909 (119886 + 119899) + 119863119906 (119899) (50)


(119886Δ120572

119909) (119899) = 119860119909 (119886 + 119899) + 119861119906 (119899)

119910 (119899) = 119862119909 (119886 + 119899) + 119863119906 (119899)(51)




1minus120572

(119911minus

1)119883(119911) = 119860119883(119911) + 119861119880(119911) Then 119883(119911) = (1(119911 minus 1)) sdot



120572119862(119868minus


(120572120572)(119860 119899minus1)119861We


(minus1) = 0




lowast119909eq)(119905) = 119891(119905 119909eq) = 0 (and (

119886Δ120572119894 119909eq)(119905) = 119891(119905 119909eq)


0




0


implies

lim119896rarrinfin

119909 (119886 + 119896ℎ) = 0 (52)




lowast



02

01

0

minus01

minus02

50 100 150 200xvalues

n steps


02

01

0

minus01

minus02

50 100 150 200yvalues

n steps


y

x

02

02

01

01

0

0

minus01

minus01

minus02

minus02




119909(119886 + 119899ℎ) =

lim119899rarrinfin

1205930120572




det(119868 minus1

119911(

119911

119911 minus 1)120572

119860ℎ120572

) = 0 (53)







(1119911)(119911(119911 minus 1))120572

119869ℎ120572

)minus1






119904)


119897 119897 = 1 119904 Let 119896

119894isin N1




119894is 119901119894


(

(

119908minus1 (120582119894) 119908minus2 (120582

119894) 119908minus119903119902 (120582

119894)

0 119908minus1 (120582119894) 119908minus119903119902+1 (120582

119894)

d

0 0 119908minus1 (120582119894)

)

)

(54)


det(119908(119860)) = det(119908(119869)) = prod119898

119894=1119908119896119894(120582

119894) and 119883(119911) is formed by




119899rarrinfin119909(119899) = lim

119911rarr1(119911 minus 1)119883(119911) = 0



are not stable



120572 for all120582 isin Spec(119860)


120572minus2120572

2120572minus2120572 ]


119890= 0 Since




1= minus2120572 minus 2120572119894

1205822







0

y

x

times107

times107

15

10

10

05

minus10

minus10

minus05


8 Conclusions




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











E(120572120573)

(120582 119911)

=

infin

sum119896=0

120582119896 (119911 + (119896 minus 1) (120572 minus 1))

(119896120572)

(119911 + 119896 (120572 minus 1))(120573minus1)

Γ (120572119896 + 120573)(7)


119864(120572120573)

(120582 119899) =

infin

sum119896=0

120582119896

120593119896120572+120573

(119899 minus 119896) (8)


(120582 119899) =

suminfin

119896=0120582119896120593119896120572+120572

(119899 minus 119896) and 119864(120572)

(120582 119899) = 119864(1205721)

(120582 119899) =

suminfin

119896=0120582119896120593119896120572+1


(120582 119899) =

119864(1)

(120582 119899) = (1 + 120582)119899



Proposition 8 Let 119864(120572120573)


Z [119864(120572120573)

(120582 sdot)] (119911) = (119911

119911 minus 1)120573

(1 minus120582

119911(

119911

119911 minus 1)120572

)

minus1

(9)



Z [119864(120572120573)

(120582 sdot)] (119911) =

infin

sum119896=0

infin

sum119899=0

120582119896

(119899 minus 119896 + 119896120572 + 120573 minus 1

119899 minus 119896) 119911minus119899

=

infin

sum119896=0

120582119896

119911minus119896

infin

sum119904=0

(119904 + 119896120572 + 120573 minus 1

119904) 119911minus119904

=

infin

sum119896=0

(120582

119911)

119896 infin

sum119904=0

(minus1)119904

(minus119896120572 minus 120573

119904) 119911minus119904

= (119911

119911 minus 1)120573 infin

sum119896=0

(120582

119911)

119896

(119911

119911 minus 1)119896120572

= (119911

119911 minus 1)120573

(1 minus120582

119911(

119911

119911 minus 1)120572

)

minus1

(10)


|120582|


(119911 minus 1)120572

= 120582119911120572minus1

(11)


119864(120572120573)

(120582 119899) = 0




120572 lt 120582 lt 0




119896120572 Z rarr R


values

120593lowast

119896120572(119899) =

(119899 minus 119896 + 119896120572

119899 minus 119896) for 119899 isin N

119896

0 for 119899 lt 119896

(12)



Z [120593lowast

119896120572] (119911) =

1

119911119896(

119911

119911 minus 1)119896120572+1

(13)

for 119911 such that |119911| gt 1


Z [120593lowast

119896120572] (119911)

=

infin

sum119899=0

120593lowast

119896120572(119899) 119911minus119899

=

infin

sum119899=119896

(119899 minus 119896 + 119896120572

119899 minus 119896) 119911minus119899

= 119911minus119896

infin

sum119904=0

(119904 + 119896120572

119904) 119911minus119904

= 119911minus119896

infin

sum119904=0

(minus1)119904

(minus119896120572 minus 1

119904) 119911minus119904

= 119911minus119896

(1 minus1

119911)minus119896120572minus1

=1

119911119896(

119911

119911 minus 1)119896120572+1

(14)



relations hold

(1) 120593lowast0120572

(119899) = 1 120593lowast119896120572

(119896) = 1 and 120593lowast1120572

(1) = 1(2) 120593lowast119896120572



(3) 120593lowast119896120572

(119899) = (1Γ(119896120572 + 1))(119899 minus 119896 + 119896120572)(119896120572) = ( 119899minus119896+119896120572119896120572

)

(4) Z[120593lowast

1120572](119911) = (1119911)(119911(119911 minus 1))

120572+1

= (1(119911 minus 1))(119911(119911 minus

1))120572

(5) Z[120593lowast119896120572

](119911) = (1119911119896) sdot (119911(119911 minus 1))119896120572+1 = (1(119911 minus 1)119896

) sdot



(119899) = ( 119899119896) and Δminus1 ( 119899minus1

119896) =

( 119899119896+1

)



119896)] (119911) =


1199112(119911 minus 1)119896+2 andZ [( 119899

119896+1)] (119911) = (1119911119896+1) sdot (119911(119911 minus 1))119896+2 =


119896)] (119911) = Z [( 119899

119896+1)] (119911) but



)] (119911) =


Z [Δminus1 ( 119899minus1119896

)] (119911) = Z [( 119899119896+1




(119860 119899 + ]) = suminfin

119896=0119860119896

120593lowast

119896120572(119899) = 119864

(120572)(119860 119899) where


(119860 119899) = sum119899

119896=0119860119896120593lowast119896120572


(120572)(119860 119899)


(119860 119899) = (119868 + 119860)119899



lowast





0one has

(Δminus120572

120593lowast

119896120572) (119899 + ]) = 120593

lowast

119896+1120572(119899) (15)



119886

Δ120572


(ℎN)119886


(119886

Δ120572

ℎlowast119909) (119905) = (


ℎ(Δℎ119909)) (119905) (16)


Note that119886

Δ120572




Δ1

ℎlowast119909)(119905) = (

119886Δ0

ℎ(Δℎ119909))(119905) = (Δ

ℎ119909)(119905) where 119905 isin (ℎN)

119886



(119886

Δ120572

ℎlowast119909) (119905) = ℎ

minus120572

(1205931minus120572

lowast Δ119909) (119899) (17)




119886The following


Δ120572


120572ℎ


Lemma 15 (see [21]) Let 119909 (ℎN)119886


Δ120572

ℎlowast119909)(119905) = ℎminus120572(

119886ℎΔ1205721lowast

119909)(119905ℎ) where 119905 isin (ℎN)119886+120572ℎ

and119909(119904) = 119909(119904ℎ)


Δ120572

lowast=

119886ℎ

Δ120572



(Δminus120572

(119886ℎ

Δ120572

lowast119909)) (119899 + 120572) = 119909 (119899 + 120572) minus 119909 (119886) (18)



(119886

Δ120572



Then

Z [119910] (119911) = ℎminus120572

(119911

119911 minus 1)1minus120572

((119911 minus 1)119883 (119911) minus 119911119909 (119886)) (19)

where 119883(119911) = Z[119909](119911) and 119909(119899) = 119909(119886 + 119899ℎ)


Δ120572

ℎlowast119909)(119905) = ℎminus120572(120593

1minus120572lowast







1the


(0Δ120572

lowast120593lowast

119896120572) (119899 + 1 minus 120572) = 120593

lowast

119896minus1120572(119899) (20)



)(119899+1minus120572) =



(119886

Δ120572

ℎlowast119909) (119899ℎ) = 119860119909 (119899ℎ + 119886) 119899 isin N

0 (21)

where 119909 (ℎZ)119886




0can be


( ]Δ120572

lowast119909) (119899) = 119860119909 (119899 + ]) 119899 isin N

0 (22a)

119909 (]) = 1199090isin R119901

(22b)




(119886

Δ120572

ℎlowast119909) (119905) = 119860119909 (119905 + 119886) 119905 isin (ℎN)

0(23a)

119909 (119886) = 1199090 1199090isin R119901 (23b)


119909 (119905) = E(120572)

(119860ℎ120572

119905

ℎ) 1199090= 119864(120572)

(119860ℎ120572

119899) 1199090

=

infin

sum119896=0

119860119896

ℎ119896120572

120593lowast

119896120572(119899) 1199090

(24)


120572minus1



(119860ℎ120572

119899) =

suminfin

119896=0119860119896ℎ119896120572120593lowast




Z [119872] (119911) =119911

119911 minus 1(119868 minus

1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

(25)


1))120572119860ℎ120572)minus1119909






Δ120572

ℎlowast119909)(119905) = 120582ℎminus120572119909(119905 + 119886) 119909(119886) = 1

Then119864(120572)


Zminus1[suminfin



120572 = 120582 into (25)




lowast





120593119896120572

(119899) =

(119899 minus 119896 + 119896120572 + 120572 minus 1

119899 minus 119896) for 119899 isin N

119896

0 for 119899 lt 119896

(26)

Proposition 19 Let 120593119896120572


Z [120593119896120572

] (119911) =1

119911119896(

119911

119911 minus 1)119896120572+120572

(27)

for 119911 such that |119911| gt 1


Z [120593119896120572

] (119911) =

infin

sum119899=0

120593119896120572

(119899) 119911minus119899

=

infin

sum119899=119896

(119899 minus 119896 + (119896 + 1) 120572 minus 1

119899 minus 119896) 119911minus119899

= 119911minus119896

infin

sum119904=0

(119904 + (119896 + 1) 120572 minus 1

119904) 119911minus119904

= 119911minus119896

infin

sum119904=0

(minus1)119904

(minus (119896 + 1) 120572

119904) 119911minus119904

= 119911minus119896

(1 minus1

119911)minus119896120572minus120572

=1

119911119896(

119911

119911 minus 1)(119896+1)120572

(28)



relations hold

(1) 1205930120572

(0) = 1 120593119896120572

(119896) = 1 and 1205931120572

(1) = 1(2) 120593119896120572

(119899) ge 0 and 120593lowast119896120572

(119899) ge 0

(3) 120593119896120572

(119899) = (119899 minus (119896 + 1) + (119896 + 1)120572)((119896+1)120572minus1)Γ((119896 + 1)120572)(4) Z[120593

0120572](119911) = (119911(119911 minus 1))120572Z[120593

1120572](119911) = (1119911)(119911(119911 minus

1))2120572(5) Z[120593

119896120572](119911) = (1119911119896)(119911(119911 minus 1))(119896+1)120572 for 119896 isin N

0

(6) For 120572 = 1 we have 1205931198961

(119899) = 120593lowast1198961

(119899) = ( 119899119896)




(119860 119899+]) = suminfin

119896=0119860119896120593119896120572

(119899) = 119864(120572120572)

(119860 119899) where ] = 120572minus

1 Note that in fact 119864(120572120572)

(119860 119899) = sum119899

119896=0119860119896120593119896120572


(120572120572)(119860 119899) always



(119860 119899) = (119868 + 119860)119899






one has

(Δminus120572

120593119896120572

) (119899 + ]) = 120593119896+1120572

(119899) (29)


)(119899 +

])](119911) = (1119911)(119911(119911 minus 1))120572 sdot Z[120593119896120572



119896120572)(119899+

])](119911) = (1119911119896+1)(119911(119911 minus 1))(119896+2)120572 = Z[120593119896+1120572




119886Δ120572


119909 (ℎN)119886


(119886Δ120572

ℎ119909) (119905) = (Δ


ℎ119909)) (119905) (30)



(119886

Δ120572

ℎlowast119909) (119905) = (

119886Δ120572

ℎ119909) (119905) minus

119909 (119886) sdot (119905 minus 119886)(minus120572)

ℎ

Γ (1 minus 120572)

= (119886Δ120572

ℎ119909) (119905) minus

119909 (119886)

ℎ120572(

119905 minus 119886

ℎminus120572

)

(31)


119886

Δ1

ℎlowast119909)(119905) =

(119886Δ1

ℎ119909)(119905) = (Δ

ℎ119909)(119905)





ℎ(119886Δ120572ℎ119909))(119905) = 119909(119905) minus 119909(119886) sdot

(ℎ1minus120572Γ(120572))119905(120572minus1)

ℎ= 119909(119905) minus 119909(119886) sdot ( 119905ℎ

120572minus1) 119905 isin (ℎN)

120572ℎ


Lemma 23 Let 119909 (ℎN)119886


ℎ119909)(119905) = ℎminus120572(

119886ℎΔ120572

1119909)(119905ℎ) where 119905 isin (ℎN)

119886+(1minus120572)ℎand

119909(119904) = 119909(119904ℎ)


Δ120572

=119886ℎ

Δ120572

1 Then the


(Δminus120572

(119886ℎ

Δ120572

119909)) (119899 + 120572) = 119909 (119899 + 120572) minus 119909 (119886) sdot (119899 + 120572

119899 + 1)

(32)



(119886Δ120572

ℎ119909)(119905) where 119905 isin (ℎN)


Then

Z [119910] (119911) = 119911 (ℎ119911

119911 minus 1)

minus120572

119883(119911) minus 119911ℎminus120572

119909 (119886) (33)

where 119883(119911) = Z[119909](119911) and 119909(119899) = 119909(119886 + 119899ℎ)


minus(1minus120572)

119909)(119899) Then Z[119910](119911) =




ℎof 119909



(0Δ120572

120593119896120572

) (119899 + 1 minus 120572) = 120593119896minus1120572

(119899) (34)

while for 119896 = 0 we get 1205930120572

(119899) = ( 119899+120572minus1119899

) = 120593120572(119899) and


1205930120572

)(119899+1minus120572) = (0Δ120572

120593120572)(119899+1minus120572) = 0


(119886Δ120572

ℎ119909) (119899ℎ) = 119860119909 (119899ℎ + 119886) 119899 isin N

0 (35)

where 119909 (ℎZ)119886


0isin R119901


( ]Δ120572

119909) (119899) = 119860119909 (119899 + ]) 119899 isin N0 (36a)

119909 (]) = 1199090isin R119901

(36b)



(119886Δ120572

ℎ119909) (119905) = 119860119909 (119905 + 119886) 119905 isin (ℎN)

0 (37a)

119909 (119886) = 1199090 1199090isin R119901

(37b)



119909 (119905) = E(120572120572)

(119860ℎ120572

119905

ℎ) 1199090= 119864(120572120572)

(119860ℎ120572

119899) 1199090

=

infin

sum119896=0

119860119896

ℎ119896120572

120593119896120572

(119899) 1199090

(38)


120572minus1




Z [119872] (119911) = (119911

119911 minus 1)120572

(119868 minus1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

(39)


minus120572





Let 119909 (ℎN)119886


(119886Δ120572






119864(120572120572)


119896=0((120582119896119911119896)(119911(119911 minus 1))119896120572+120572)](119899)





119886Δ120572


119886rarr

R is defined by

(119886Δ120572

ℎ119909) (119905) =

(119905minus119886)ℎ

sum119904=0

119886(120572)

119904119909 (119905 minus 119904ℎ) (40)

where

119886(120572)

119904= (minus1)

119904

(120572

119904)(

1

ℎ120572) with

(120572

119904) =

1 for 119904 = 0


119904 for 119904 isin N

(41)


(119886Δ120572

ℎ119909)(119905) where 119905 isin (ℎN)

119886and 119905 = 119886 + 119899ℎ 119899 isin N

0 Then

Z [119910] (119911) = (ℎ119911

119911 minus 1)

minus120572

119883 (119911) (42)

where 119883(119911) = Z[119909](119911) and 119909(119899) = 119909(119886 + 119899ℎ)


ℎ119909](119911) = Z[119886(120572)

119904lowast119909][119911] =

Z[119886(120572)119904




ℎ119909)(119899ℎ) =

(0Δ120572


0 or 119909(119905) =




ℎ119910)(119905 + ℎ) =

(119886Δ120572


119886

Proof Let119891(119899) = (0Δ120572



ℎ119910)(0) = 119911(ℎ119911(119911minus


Z[119892] where 119883(119911) = Z[119909](119911) 119909(119899) = 119910(119899ℎ) = 119909(119886 + 119899ℎ) and119892(119899) = (

119886Δ120572

ℎ119909)(119905) 119905 isin (ℎN)




0Δ120572

ℎ119910)(119899ℎ + ℎ) as for the




(0Δ120572

119910) (119905 + ℎ) = 119860119910 (119905) 119905 isin (ℎN)0

119910 (0) = 1199100 1199100isin R119901

(43)




(119860ℎ120572 + 119868120572)Φ(119905) with Φ(0) = 119868



(0Δ120572

ℎ119910) (119905 + ℎ) = 119860119910 (119905) 119905 isin (ℎN)

0

119910 (0) = 1199100 1199100isin R119901

(44)


119910 (119905) = 119864(120572120572)

(119860ℎ120572

119905

ℎ) 1199100 (45)




Φ (119899ℎ) = Zminus1

[(119911

119911 minus 1)120572

(119868 minus1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

] (119899)

(46)




ℎminus119886

Δ120572

ℎlowast)119909)(119905) where 119905 =

119886 + (1 minus 120572)ℎ + 119899ℎ Then

Z [119910] (119911) = ℎminus120572

119911((119911

119911 minus 1)1minus120572

minus 1)119909 (119886) (47)



)] (119911)119909(119886) =


119904=0(minus1)119904

( 120572minus1119904


1)119909(119886)



119864(120572120572)

(119860 119899) = (119864(120572)


) (119899) (48)


Z [119864(120572)


] (119911)

= Z [119864(120572)

(119860 sdot)] (119911)Z [120593120572minus1

] (119911)

=119911

119911 minus 1(119868 minus

1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

(119911

119911 minus 1)120572minus1

= (119911

119911 minus 1)120572

(119868 minus1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

(49)



](119911) =

Z[119864(120572120572)





(119886Δ120572

lowast119909) (119899) = 119860119909 (119886 + 119899) + 119861119906 (119899)

119910 (119899) = 119862119909 (119886 + 119899) + 119863119906 (119899) (50)


(119886Δ120572

119909) (119899) = 119860119909 (119886 + 119899) + 119861119906 (119899)

119910 (119899) = 119862119909 (119886 + 119899) + 119863119906 (119899)(51)




1minus120572

(119911minus

1)119883(119911) = 119860119883(119911) + 119861119880(119911) Then 119883(119911) = (1(119911 minus 1)) sdot



120572119862(119868minus


(120572120572)(119860 119899minus1)119861We


(minus1) = 0




lowast119909eq)(119905) = 119891(119905 119909eq) = 0 (and (

119886Δ120572119894 119909eq)(119905) = 119891(119905 119909eq)


0




0


implies

lim119896rarrinfin

119909 (119886 + 119896ℎ) = 0 (52)




lowast



02

01

0

minus01

minus02

50 100 150 200xvalues

n steps


02

01

0

minus01

minus02

50 100 150 200yvalues

n steps


y

x

02

02

01

01

0

0

minus01

minus01

minus02

minus02




119909(119886 + 119899ℎ) =

lim119899rarrinfin

1205930120572




det(119868 minus1

119911(

119911

119911 minus 1)120572

119860ℎ120572

) = 0 (53)







(1119911)(119911(119911 minus 1))120572

119869ℎ120572

)minus1






119904)


119897 119897 = 1 119904 Let 119896

119894isin N1




119894is 119901119894


(

(

119908minus1 (120582119894) 119908minus2 (120582

119894) 119908minus119903119902 (120582

119894)

0 119908minus1 (120582119894) 119908minus119903119902+1 (120582

119894)

d

0 0 119908minus1 (120582119894)

)

)

(54)


det(119908(119860)) = det(119908(119869)) = prod119898

119894=1119908119896119894(120582

119894) and 119883(119911) is formed by




119899rarrinfin119909(119899) = lim

119911rarr1(119911 minus 1)119883(119911) = 0



are not stable



120572 for all120582 isin Spec(119860)


120572minus2120572

2120572minus2120572 ]


119890= 0 Since




1= minus2120572 minus 2120572119894

1205822







0

y

x

times107

times107

15

10

10

05

minus10

minus10

minus05


8 Conclusions




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(3) 120593lowast119896120572

(119899) = (1Γ(119896120572 + 1))(119899 minus 119896 + 119896120572)(119896120572) = ( 119899minus119896+119896120572119896120572

)

(4) Z[120593lowast

1120572](119911) = (1119911)(119911(119911 minus 1))

120572+1

= (1(119911 minus 1))(119911(119911 minus

1))120572

(5) Z[120593lowast119896120572

](119911) = (1119911119896) sdot (119911(119911 minus 1))119896120572+1 = (1(119911 minus 1)119896

) sdot



(119899) = ( 119899119896) and Δminus1 ( 119899minus1

119896) =

( 119899119896+1

)



119896)] (119911) =


1199112(119911 minus 1)119896+2 andZ [( 119899

119896+1)] (119911) = (1119911119896+1) sdot (119911(119911 minus 1))119896+2 =


119896)] (119911) = Z [( 119899

119896+1)] (119911) but



)] (119911) =


Z [Δminus1 ( 119899minus1119896

)] (119911) = Z [( 119899119896+1




(119860 119899 + ]) = suminfin

119896=0119860119896

120593lowast

119896120572(119899) = 119864

(120572)(119860 119899) where


(119860 119899) = sum119899

119896=0119860119896120593lowast119896120572


(120572)(119860 119899)


(119860 119899) = (119868 + 119860)119899



lowast





0one has

(Δminus120572

120593lowast

119896120572) (119899 + ]) = 120593

lowast

119896+1120572(119899) (15)



119886

Δ120572


(ℎN)119886


(119886

Δ120572

ℎlowast119909) (119905) = (


ℎ(Δℎ119909)) (119905) (16)


Note that119886

Δ120572




Δ1

ℎlowast119909)(119905) = (

119886Δ0

ℎ(Δℎ119909))(119905) = (Δ

ℎ119909)(119905) where 119905 isin (ℎN)

119886



(119886

Δ120572

ℎlowast119909) (119905) = ℎ

minus120572

(1205931minus120572

lowast Δ119909) (119899) (17)




119886The following


Δ120572


120572ℎ


Lemma 15 (see [21]) Let 119909 (ℎN)119886


Δ120572

ℎlowast119909)(119905) = ℎminus120572(

119886ℎΔ1205721lowast

119909)(119905ℎ) where 119905 isin (ℎN)119886+120572ℎ

and119909(119904) = 119909(119904ℎ)


Δ120572

lowast=

119886ℎ

Δ120572



(Δminus120572

(119886ℎ

Δ120572

lowast119909)) (119899 + 120572) = 119909 (119899 + 120572) minus 119909 (119886) (18)



(119886

Δ120572



Then

Z [119910] (119911) = ℎminus120572

(119911

119911 minus 1)1minus120572

((119911 minus 1)119883 (119911) minus 119911119909 (119886)) (19)

where 119883(119911) = Z[119909](119911) and 119909(119899) = 119909(119886 + 119899ℎ)


Δ120572

ℎlowast119909)(119905) = ℎminus120572(120593

1minus120572lowast







1the


(0Δ120572

lowast120593lowast

119896120572) (119899 + 1 minus 120572) = 120593

lowast

119896minus1120572(119899) (20)



)(119899+1minus120572) =



(119886

Δ120572

ℎlowast119909) (119899ℎ) = 119860119909 (119899ℎ + 119886) 119899 isin N

0 (21)

where 119909 (ℎZ)119886




0can be


( ]Δ120572

lowast119909) (119899) = 119860119909 (119899 + ]) 119899 isin N

0 (22a)

119909 (]) = 1199090isin R119901

(22b)




(119886

Δ120572

ℎlowast119909) (119905) = 119860119909 (119905 + 119886) 119905 isin (ℎN)

0(23a)

119909 (119886) = 1199090 1199090isin R119901 (23b)


119909 (119905) = E(120572)

(119860ℎ120572

119905

ℎ) 1199090= 119864(120572)

(119860ℎ120572

119899) 1199090

=

infin

sum119896=0

119860119896

ℎ119896120572

120593lowast

119896120572(119899) 1199090

(24)


120572minus1



(119860ℎ120572

119899) =

suminfin

119896=0119860119896ℎ119896120572120593lowast




Z [119872] (119911) =119911

119911 minus 1(119868 minus

1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

(25)


1))120572119860ℎ120572)minus1119909






Δ120572

ℎlowast119909)(119905) = 120582ℎminus120572119909(119905 + 119886) 119909(119886) = 1

Then119864(120572)


Zminus1[suminfin



120572 = 120582 into (25)




lowast





120593119896120572

(119899) =

(119899 minus 119896 + 119896120572 + 120572 minus 1

119899 minus 119896) for 119899 isin N

119896

0 for 119899 lt 119896

(26)

Proposition 19 Let 120593119896120572


Z [120593119896120572

] (119911) =1

119911119896(

119911

119911 minus 1)119896120572+120572

(27)

for 119911 such that |119911| gt 1


Z [120593119896120572

] (119911) =

infin

sum119899=0

120593119896120572

(119899) 119911minus119899

=

infin

sum119899=119896

(119899 minus 119896 + (119896 + 1) 120572 minus 1

119899 minus 119896) 119911minus119899

= 119911minus119896

infin

sum119904=0

(119904 + (119896 + 1) 120572 minus 1

119904) 119911minus119904

= 119911minus119896

infin

sum119904=0

(minus1)119904

(minus (119896 + 1) 120572

119904) 119911minus119904

= 119911minus119896

(1 minus1

119911)minus119896120572minus120572

=1

119911119896(

119911

119911 minus 1)(119896+1)120572

(28)



relations hold

(1) 1205930120572

(0) = 1 120593119896120572

(119896) = 1 and 1205931120572

(1) = 1(2) 120593119896120572

(119899) ge 0 and 120593lowast119896120572

(119899) ge 0

(3) 120593119896120572

(119899) = (119899 minus (119896 + 1) + (119896 + 1)120572)((119896+1)120572minus1)Γ((119896 + 1)120572)(4) Z[120593

0120572](119911) = (119911(119911 minus 1))120572Z[120593

1120572](119911) = (1119911)(119911(119911 minus

1))2120572(5) Z[120593

119896120572](119911) = (1119911119896)(119911(119911 minus 1))(119896+1)120572 for 119896 isin N

0

(6) For 120572 = 1 we have 1205931198961

(119899) = 120593lowast1198961

(119899) = ( 119899119896)




(119860 119899+]) = suminfin

119896=0119860119896120593119896120572

(119899) = 119864(120572120572)

(119860 119899) where ] = 120572minus

1 Note that in fact 119864(120572120572)

(119860 119899) = sum119899

119896=0119860119896120593119896120572


(120572120572)(119860 119899) always



(119860 119899) = (119868 + 119860)119899






one has

(Δminus120572

120593119896120572

) (119899 + ]) = 120593119896+1120572

(119899) (29)


)(119899 +

])](119911) = (1119911)(119911(119911 minus 1))120572 sdot Z[120593119896120572



119896120572)(119899+

])](119911) = (1119911119896+1)(119911(119911 minus 1))(119896+2)120572 = Z[120593119896+1120572




119886Δ120572


119909 (ℎN)119886


(119886Δ120572

ℎ119909) (119905) = (Δ


ℎ119909)) (119905) (30)



(119886

Δ120572

ℎlowast119909) (119905) = (

119886Δ120572

ℎ119909) (119905) minus

119909 (119886) sdot (119905 minus 119886)(minus120572)

ℎ

Γ (1 minus 120572)

= (119886Δ120572

ℎ119909) (119905) minus

119909 (119886)

ℎ120572(

119905 minus 119886

ℎminus120572

)

(31)


119886

Δ1

ℎlowast119909)(119905) =

(119886Δ1

ℎ119909)(119905) = (Δ

ℎ119909)(119905)





ℎ(119886Δ120572ℎ119909))(119905) = 119909(119905) minus 119909(119886) sdot

(ℎ1minus120572Γ(120572))119905(120572minus1)

ℎ= 119909(119905) minus 119909(119886) sdot ( 119905ℎ

120572minus1) 119905 isin (ℎN)

120572ℎ


Lemma 23 Let 119909 (ℎN)119886


ℎ119909)(119905) = ℎminus120572(

119886ℎΔ120572

1119909)(119905ℎ) where 119905 isin (ℎN)

119886+(1minus120572)ℎand

119909(119904) = 119909(119904ℎ)


Δ120572

=119886ℎ

Δ120572

1 Then the


(Δminus120572

(119886ℎ

Δ120572

119909)) (119899 + 120572) = 119909 (119899 + 120572) minus 119909 (119886) sdot (119899 + 120572

119899 + 1)

(32)



(119886Δ120572

ℎ119909)(119905) where 119905 isin (ℎN)


Then

Z [119910] (119911) = 119911 (ℎ119911

119911 minus 1)

minus120572

119883(119911) minus 119911ℎminus120572

119909 (119886) (33)

where 119883(119911) = Z[119909](119911) and 119909(119899) = 119909(119886 + 119899ℎ)


minus(1minus120572)

119909)(119899) Then Z[119910](119911) =




ℎof 119909



(0Δ120572

120593119896120572

) (119899 + 1 minus 120572) = 120593119896minus1120572

(119899) (34)

while for 119896 = 0 we get 1205930120572

(119899) = ( 119899+120572minus1119899

) = 120593120572(119899) and


1205930120572

)(119899+1minus120572) = (0Δ120572

120593120572)(119899+1minus120572) = 0


(119886Δ120572

ℎ119909) (119899ℎ) = 119860119909 (119899ℎ + 119886) 119899 isin N

0 (35)

where 119909 (ℎZ)119886


0isin R119901


( ]Δ120572

119909) (119899) = 119860119909 (119899 + ]) 119899 isin N0 (36a)

119909 (]) = 1199090isin R119901

(36b)



(119886Δ120572

ℎ119909) (119905) = 119860119909 (119905 + 119886) 119905 isin (ℎN)

0 (37a)

119909 (119886) = 1199090 1199090isin R119901

(37b)



119909 (119905) = E(120572120572)

(119860ℎ120572

119905

ℎ) 1199090= 119864(120572120572)

(119860ℎ120572

119899) 1199090

=

infin

sum119896=0

119860119896

ℎ119896120572

120593119896120572

(119899) 1199090

(38)


120572minus1




Z [119872] (119911) = (119911

119911 minus 1)120572

(119868 minus1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

(39)


minus120572





Let 119909 (ℎN)119886


(119886Δ120572






119864(120572120572)


119896=0((120582119896119911119896)(119911(119911 minus 1))119896120572+120572)](119899)





119886Δ120572


119886rarr

R is defined by

(119886Δ120572

ℎ119909) (119905) =

(119905minus119886)ℎ

sum119904=0

119886(120572)

119904119909 (119905 minus 119904ℎ) (40)

where

119886(120572)

119904= (minus1)

119904

(120572

119904)(

1

ℎ120572) with

(120572

119904) =

1 for 119904 = 0


119904 for 119904 isin N

(41)


(119886Δ120572

ℎ119909)(119905) where 119905 isin (ℎN)

119886and 119905 = 119886 + 119899ℎ 119899 isin N

0 Then

Z [119910] (119911) = (ℎ119911

119911 minus 1)

minus120572

119883 (119911) (42)

where 119883(119911) = Z[119909](119911) and 119909(119899) = 119909(119886 + 119899ℎ)


ℎ119909](119911) = Z[119886(120572)

119904lowast119909][119911] =

Z[119886(120572)119904




ℎ119909)(119899ℎ) =

(0Δ120572


0 or 119909(119905) =




ℎ119910)(119905 + ℎ) =

(119886Δ120572


119886

Proof Let119891(119899) = (0Δ120572



ℎ119910)(0) = 119911(ℎ119911(119911minus


Z[119892] where 119883(119911) = Z[119909](119911) 119909(119899) = 119910(119899ℎ) = 119909(119886 + 119899ℎ) and119892(119899) = (

119886Δ120572

ℎ119909)(119905) 119905 isin (ℎN)




0Δ120572

ℎ119910)(119899ℎ + ℎ) as for the




(0Δ120572

119910) (119905 + ℎ) = 119860119910 (119905) 119905 isin (ℎN)0

119910 (0) = 1199100 1199100isin R119901

(43)




(119860ℎ120572 + 119868120572)Φ(119905) with Φ(0) = 119868



(0Δ120572

ℎ119910) (119905 + ℎ) = 119860119910 (119905) 119905 isin (ℎN)

0

119910 (0) = 1199100 1199100isin R119901

(44)


119910 (119905) = 119864(120572120572)

(119860ℎ120572

119905

ℎ) 1199100 (45)




Φ (119899ℎ) = Zminus1

[(119911

119911 minus 1)120572

(119868 minus1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

] (119899)

(46)




ℎminus119886

Δ120572

ℎlowast)119909)(119905) where 119905 =

119886 + (1 minus 120572)ℎ + 119899ℎ Then

Z [119910] (119911) = ℎminus120572

119911((119911

119911 minus 1)1minus120572

minus 1)119909 (119886) (47)



)] (119911)119909(119886) =


119904=0(minus1)119904

( 120572minus1119904


1)119909(119886)



119864(120572120572)

(119860 119899) = (119864(120572)


) (119899) (48)


Z [119864(120572)


] (119911)

= Z [119864(120572)

(119860 sdot)] (119911)Z [120593120572minus1

] (119911)

=119911

119911 minus 1(119868 minus

1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

(119911

119911 minus 1)120572minus1

= (119911

119911 minus 1)120572

(119868 minus1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

(49)



](119911) =

Z[119864(120572120572)





(119886Δ120572

lowast119909) (119899) = 119860119909 (119886 + 119899) + 119861119906 (119899)

119910 (119899) = 119862119909 (119886 + 119899) + 119863119906 (119899) (50)


(119886Δ120572

119909) (119899) = 119860119909 (119886 + 119899) + 119861119906 (119899)

119910 (119899) = 119862119909 (119886 + 119899) + 119863119906 (119899)(51)




1minus120572

(119911minus

1)119883(119911) = 119860119883(119911) + 119861119880(119911) Then 119883(119911) = (1(119911 minus 1)) sdot



120572119862(119868minus


(120572120572)(119860 119899minus1)119861We


(minus1) = 0




lowast119909eq)(119905) = 119891(119905 119909eq) = 0 (and (

119886Δ120572119894 119909eq)(119905) = 119891(119905 119909eq)


0




0


implies

lim119896rarrinfin

119909 (119886 + 119896ℎ) = 0 (52)




lowast



02

01

0

minus01

minus02

50 100 150 200xvalues

n steps


02

01

0

minus01

minus02

50 100 150 200yvalues

n steps


y

x

02

02

01

01

0

0

minus01

minus01

minus02

minus02




119909(119886 + 119899ℎ) =

lim119899rarrinfin

1205930120572




det(119868 minus1

119911(

119911

119911 minus 1)120572

119860ℎ120572

) = 0 (53)







(1119911)(119911(119911 minus 1))120572

119869ℎ120572

)minus1






119904)


119897 119897 = 1 119904 Let 119896

119894isin N1




119894is 119901119894


(

(

119908minus1 (120582119894) 119908minus2 (120582

119894) 119908minus119903119902 (120582

119894)

0 119908minus1 (120582119894) 119908minus119903119902+1 (120582

119894)

d

0 0 119908minus1 (120582119894)

)

)

(54)


det(119908(119860)) = det(119908(119869)) = prod119898

119894=1119908119896119894(120582

119894) and 119883(119911) is formed by




119899rarrinfin119909(119899) = lim

119911rarr1(119911 minus 1)119883(119911) = 0



are not stable



120572 for all120582 isin Spec(119860)


120572minus2120572

2120572minus2120572 ]


119890= 0 Since




1= minus2120572 minus 2120572119894

1205822







0

y

x

times107

times107

15

10

10

05

minus10

minus10

minus05


8 Conclusions




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(119886

Δ120572

ℎlowast119909) (119905) = 119860119909 (119905 + 119886) 119905 isin (ℎN)

0(23a)

119909 (119886) = 1199090 1199090isin R119901 (23b)


119909 (119905) = E(120572)

(119860ℎ120572

119905

ℎ) 1199090= 119864(120572)

(119860ℎ120572

119899) 1199090

=

infin

sum119896=0

119860119896

ℎ119896120572

120593lowast

119896120572(119899) 1199090

(24)


120572minus1



(119860ℎ120572

119899) =

suminfin

119896=0119860119896ℎ119896120572120593lowast




Z [119872] (119911) =119911

119911 minus 1(119868 minus

1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

(25)


1))120572119860ℎ120572)minus1119909






Δ120572

ℎlowast119909)(119905) = 120582ℎminus120572119909(119905 + 119886) 119909(119886) = 1

Then119864(120572)


Zminus1[suminfin



120572 = 120582 into (25)




lowast





120593119896120572

(119899) =

(119899 minus 119896 + 119896120572 + 120572 minus 1

119899 minus 119896) for 119899 isin N

119896

0 for 119899 lt 119896

(26)

Proposition 19 Let 120593119896120572


Z [120593119896120572

] (119911) =1

119911119896(

119911

119911 minus 1)119896120572+120572

(27)

for 119911 such that |119911| gt 1


Z [120593119896120572

] (119911) =

infin

sum119899=0

120593119896120572

(119899) 119911minus119899

=

infin

sum119899=119896

(119899 minus 119896 + (119896 + 1) 120572 minus 1

119899 minus 119896) 119911minus119899

= 119911minus119896

infin

sum119904=0

(119904 + (119896 + 1) 120572 minus 1

119904) 119911minus119904

= 119911minus119896

infin

sum119904=0

(minus1)119904

(minus (119896 + 1) 120572

119904) 119911minus119904

= 119911minus119896

(1 minus1

119911)minus119896120572minus120572

=1

119911119896(

119911

119911 minus 1)(119896+1)120572

(28)



relations hold

(1) 1205930120572

(0) = 1 120593119896120572

(119896) = 1 and 1205931120572

(1) = 1(2) 120593119896120572

(119899) ge 0 and 120593lowast119896120572

(119899) ge 0

(3) 120593119896120572

(119899) = (119899 minus (119896 + 1) + (119896 + 1)120572)((119896+1)120572minus1)Γ((119896 + 1)120572)(4) Z[120593

0120572](119911) = (119911(119911 minus 1))120572Z[120593

1120572](119911) = (1119911)(119911(119911 minus

1))2120572(5) Z[120593

119896120572](119911) = (1119911119896)(119911(119911 minus 1))(119896+1)120572 for 119896 isin N

0

(6) For 120572 = 1 we have 1205931198961

(119899) = 120593lowast1198961

(119899) = ( 119899119896)




(119860 119899+]) = suminfin

119896=0119860119896120593119896120572

(119899) = 119864(120572120572)

(119860 119899) where ] = 120572minus

1 Note that in fact 119864(120572120572)

(119860 119899) = sum119899

119896=0119860119896120593119896120572


(120572120572)(119860 119899) always



(119860 119899) = (119868 + 119860)119899






one has

(Δminus120572

120593119896120572

) (119899 + ]) = 120593119896+1120572

(119899) (29)


)(119899 +

])](119911) = (1119911)(119911(119911 minus 1))120572 sdot Z[120593119896120572



119896120572)(119899+

])](119911) = (1119911119896+1)(119911(119911 minus 1))(119896+2)120572 = Z[120593119896+1120572




119886Δ120572


119909 (ℎN)119886


(119886Δ120572

ℎ119909) (119905) = (Δ


ℎ119909)) (119905) (30)



(119886

Δ120572

ℎlowast119909) (119905) = (

119886Δ120572

ℎ119909) (119905) minus

119909 (119886) sdot (119905 minus 119886)(minus120572)

ℎ

Γ (1 minus 120572)

= (119886Δ120572

ℎ119909) (119905) minus

119909 (119886)

ℎ120572(

119905 minus 119886

ℎminus120572

)

(31)


119886

Δ1

ℎlowast119909)(119905) =

(119886Δ1

ℎ119909)(119905) = (Δ

ℎ119909)(119905)





ℎ(119886Δ120572ℎ119909))(119905) = 119909(119905) minus 119909(119886) sdot

(ℎ1minus120572Γ(120572))119905(120572minus1)

ℎ= 119909(119905) minus 119909(119886) sdot ( 119905ℎ

120572minus1) 119905 isin (ℎN)

120572ℎ


Lemma 23 Let 119909 (ℎN)119886


ℎ119909)(119905) = ℎminus120572(

119886ℎΔ120572

1119909)(119905ℎ) where 119905 isin (ℎN)

119886+(1minus120572)ℎand

119909(119904) = 119909(119904ℎ)


Δ120572

=119886ℎ

Δ120572

1 Then the


(Δminus120572

(119886ℎ

Δ120572

119909)) (119899 + 120572) = 119909 (119899 + 120572) minus 119909 (119886) sdot (119899 + 120572

119899 + 1)

(32)



(119886Δ120572

ℎ119909)(119905) where 119905 isin (ℎN)


Then

Z [119910] (119911) = 119911 (ℎ119911

119911 minus 1)

minus120572

119883(119911) minus 119911ℎminus120572

119909 (119886) (33)

where 119883(119911) = Z[119909](119911) and 119909(119899) = 119909(119886 + 119899ℎ)


minus(1minus120572)

119909)(119899) Then Z[119910](119911) =




ℎof 119909



(0Δ120572

120593119896120572

) (119899 + 1 minus 120572) = 120593119896minus1120572

(119899) (34)

while for 119896 = 0 we get 1205930120572

(119899) = ( 119899+120572minus1119899

) = 120593120572(119899) and


1205930120572

)(119899+1minus120572) = (0Δ120572

120593120572)(119899+1minus120572) = 0


(119886Δ120572

ℎ119909) (119899ℎ) = 119860119909 (119899ℎ + 119886) 119899 isin N

0 (35)

where 119909 (ℎZ)119886


0isin R119901


( ]Δ120572

119909) (119899) = 119860119909 (119899 + ]) 119899 isin N0 (36a)

119909 (]) = 1199090isin R119901

(36b)



(119886Δ120572

ℎ119909) (119905) = 119860119909 (119905 + 119886) 119905 isin (ℎN)

0 (37a)

119909 (119886) = 1199090 1199090isin R119901

(37b)



119909 (119905) = E(120572120572)

(119860ℎ120572

119905

ℎ) 1199090= 119864(120572120572)

(119860ℎ120572

119899) 1199090

=

infin

sum119896=0

119860119896

ℎ119896120572

120593119896120572

(119899) 1199090

(38)


120572minus1




Z [119872] (119911) = (119911

119911 minus 1)120572

(119868 minus1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

(39)


minus120572





Let 119909 (ℎN)119886


(119886Δ120572






119864(120572120572)


119896=0((120582119896119911119896)(119911(119911 minus 1))119896120572+120572)](119899)





119886Δ120572


119886rarr

R is defined by

(119886Δ120572

ℎ119909) (119905) =

(119905minus119886)ℎ

sum119904=0

119886(120572)

119904119909 (119905 minus 119904ℎ) (40)

where

119886(120572)

119904= (minus1)

119904

(120572

119904)(

1

ℎ120572) with

(120572

119904) =

1 for 119904 = 0


119904 for 119904 isin N

(41)


(119886Δ120572

ℎ119909)(119905) where 119905 isin (ℎN)

119886and 119905 = 119886 + 119899ℎ 119899 isin N

0 Then

Z [119910] (119911) = (ℎ119911

119911 minus 1)

minus120572

119883 (119911) (42)

where 119883(119911) = Z[119909](119911) and 119909(119899) = 119909(119886 + 119899ℎ)


ℎ119909](119911) = Z[119886(120572)

119904lowast119909][119911] =

Z[119886(120572)119904




ℎ119909)(119899ℎ) =

(0Δ120572


0 or 119909(119905) =




ℎ119910)(119905 + ℎ) =

(119886Δ120572


119886

Proof Let119891(119899) = (0Δ120572



ℎ119910)(0) = 119911(ℎ119911(119911minus


Z[119892] where 119883(119911) = Z[119909](119911) 119909(119899) = 119910(119899ℎ) = 119909(119886 + 119899ℎ) and119892(119899) = (

119886Δ120572

ℎ119909)(119905) 119905 isin (ℎN)




0Δ120572

ℎ119910)(119899ℎ + ℎ) as for the




(0Δ120572

119910) (119905 + ℎ) = 119860119910 (119905) 119905 isin (ℎN)0

119910 (0) = 1199100 1199100isin R119901

(43)




(119860ℎ120572 + 119868120572)Φ(119905) with Φ(0) = 119868



(0Δ120572

ℎ119910) (119905 + ℎ) = 119860119910 (119905) 119905 isin (ℎN)

0

119910 (0) = 1199100 1199100isin R119901

(44)


119910 (119905) = 119864(120572120572)

(119860ℎ120572

119905

ℎ) 1199100 (45)




Φ (119899ℎ) = Zminus1

[(119911

119911 minus 1)120572

(119868 minus1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

] (119899)

(46)




ℎminus119886

Δ120572

ℎlowast)119909)(119905) where 119905 =

119886 + (1 minus 120572)ℎ + 119899ℎ Then

Z [119910] (119911) = ℎminus120572

119911((119911

119911 minus 1)1minus120572

minus 1)119909 (119886) (47)



)] (119911)119909(119886) =


119904=0(minus1)119904

( 120572minus1119904


1)119909(119886)



119864(120572120572)

(119860 119899) = (119864(120572)


) (119899) (48)


Z [119864(120572)


] (119911)

= Z [119864(120572)

(119860 sdot)] (119911)Z [120593120572minus1

] (119911)

=119911

119911 minus 1(119868 minus

1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

(119911

119911 minus 1)120572minus1

= (119911

119911 minus 1)120572

(119868 minus1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

(49)



](119911) =

Z[119864(120572120572)





(119886Δ120572

lowast119909) (119899) = 119860119909 (119886 + 119899) + 119861119906 (119899)

119910 (119899) = 119862119909 (119886 + 119899) + 119863119906 (119899) (50)


(119886Δ120572

119909) (119899) = 119860119909 (119886 + 119899) + 119861119906 (119899)

119910 (119899) = 119862119909 (119886 + 119899) + 119863119906 (119899)(51)




1minus120572

(119911minus

1)119883(119911) = 119860119883(119911) + 119861119880(119911) Then 119883(119911) = (1(119911 minus 1)) sdot



120572119862(119868minus


(120572120572)(119860 119899minus1)119861We


(minus1) = 0




lowast119909eq)(119905) = 119891(119905 119909eq) = 0 (and (

119886Δ120572119894 119909eq)(119905) = 119891(119905 119909eq)


0




0


implies

lim119896rarrinfin

119909 (119886 + 119896ℎ) = 0 (52)




lowast



02

01

0

minus01

minus02

50 100 150 200xvalues

n steps


02

01

0

minus01

minus02

50 100 150 200yvalues

n steps


y

x

02

02

01

01

0

0

minus01

minus01

minus02

minus02




119909(119886 + 119899ℎ) =

lim119899rarrinfin

1205930120572




det(119868 minus1

119911(

119911

119911 minus 1)120572

119860ℎ120572

) = 0 (53)







(1119911)(119911(119911 minus 1))120572

119869ℎ120572

)minus1






119904)


119897 119897 = 1 119904 Let 119896

119894isin N1




119894is 119901119894


(

(

119908minus1 (120582119894) 119908minus2 (120582

119894) 119908minus119903119902 (120582

119894)

0 119908minus1 (120582119894) 119908minus119903119902+1 (120582

119894)

d

0 0 119908minus1 (120582119894)

)

)

(54)


det(119908(119860)) = det(119908(119869)) = prod119898

119894=1119908119896119894(120582

119894) and 119883(119911) is formed by




119899rarrinfin119909(119899) = lim

119911rarr1(119911 minus 1)119883(119911) = 0



are not stable



120572 for all120582 isin Spec(119860)


120572minus2120572

2120572minus2120572 ]


119890= 0 Since




1= minus2120572 minus 2120572119894

1205822







0

y

x

times107

times107

15

10

10

05

minus10

minus10

minus05


8 Conclusions




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(119860 119899) = (119868 + 119860)119899






one has

(Δminus120572

120593119896120572

) (119899 + ]) = 120593119896+1120572

(119899) (29)


)(119899 +

])](119911) = (1119911)(119911(119911 minus 1))120572 sdot Z[120593119896120572



119896120572)(119899+

])](119911) = (1119911119896+1)(119911(119911 minus 1))(119896+2)120572 = Z[120593119896+1120572




119886Δ120572


119909 (ℎN)119886


(119886Δ120572

ℎ119909) (119905) = (Δ


ℎ119909)) (119905) (30)



(119886

Δ120572

ℎlowast119909) (119905) = (

119886Δ120572

ℎ119909) (119905) minus

119909 (119886) sdot (119905 minus 119886)(minus120572)

ℎ

Γ (1 minus 120572)

= (119886Δ120572

ℎ119909) (119905) minus

119909 (119886)

ℎ120572(

119905 minus 119886

ℎminus120572

)

(31)


119886

Δ1

ℎlowast119909)(119905) =

(119886Δ1

ℎ119909)(119905) = (Δ

ℎ119909)(119905)





ℎ(119886Δ120572ℎ119909))(119905) = 119909(119905) minus 119909(119886) sdot

(ℎ1minus120572Γ(120572))119905(120572minus1)

ℎ= 119909(119905) minus 119909(119886) sdot ( 119905ℎ

120572minus1) 119905 isin (ℎN)

120572ℎ


Lemma 23 Let 119909 (ℎN)119886


ℎ119909)(119905) = ℎminus120572(

119886ℎΔ120572

1119909)(119905ℎ) where 119905 isin (ℎN)

119886+(1minus120572)ℎand

119909(119904) = 119909(119904ℎ)


Δ120572

=119886ℎ

Δ120572

1 Then the


(Δminus120572

(119886ℎ

Δ120572

119909)) (119899 + 120572) = 119909 (119899 + 120572) minus 119909 (119886) sdot (119899 + 120572

119899 + 1)

(32)



(119886Δ120572

ℎ119909)(119905) where 119905 isin (ℎN)


Then

Z [119910] (119911) = 119911 (ℎ119911

119911 minus 1)

minus120572

119883(119911) minus 119911ℎminus120572

119909 (119886) (33)

where 119883(119911) = Z[119909](119911) and 119909(119899) = 119909(119886 + 119899ℎ)


minus(1minus120572)

119909)(119899) Then Z[119910](119911) =




ℎof 119909



(0Δ120572

120593119896120572

) (119899 + 1 minus 120572) = 120593119896minus1120572

(119899) (34)

while for 119896 = 0 we get 1205930120572

(119899) = ( 119899+120572minus1119899

) = 120593120572(119899) and


1205930120572

)(119899+1minus120572) = (0Δ120572

120593120572)(119899+1minus120572) = 0


(119886Δ120572

ℎ119909) (119899ℎ) = 119860119909 (119899ℎ + 119886) 119899 isin N

0 (35)

where 119909 (ℎZ)119886


0isin R119901


( ]Δ120572

119909) (119899) = 119860119909 (119899 + ]) 119899 isin N0 (36a)

119909 (]) = 1199090isin R119901

(36b)



(119886Δ120572

ℎ119909) (119905) = 119860119909 (119905 + 119886) 119905 isin (ℎN)

0 (37a)

119909 (119886) = 1199090 1199090isin R119901

(37b)



119909 (119905) = E(120572120572)

(119860ℎ120572

119905

ℎ) 1199090= 119864(120572120572)

(119860ℎ120572

119899) 1199090

=

infin

sum119896=0

119860119896

ℎ119896120572

120593119896120572

(119899) 1199090

(38)


120572minus1




Z [119872] (119911) = (119911

119911 minus 1)120572

(119868 minus1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

(39)


minus120572





Let 119909 (ℎN)119886


(119886Δ120572






119864(120572120572)


119896=0((120582119896119911119896)(119911(119911 minus 1))119896120572+120572)](119899)





119886Δ120572


119886rarr

R is defined by

(119886Δ120572

ℎ119909) (119905) =

(119905minus119886)ℎ

sum119904=0

119886(120572)

119904119909 (119905 minus 119904ℎ) (40)

where

119886(120572)

119904= (minus1)

119904

(120572

119904)(

1

ℎ120572) with

(120572

119904) =

1 for 119904 = 0


119904 for 119904 isin N

(41)


(119886Δ120572

ℎ119909)(119905) where 119905 isin (ℎN)

119886and 119905 = 119886 + 119899ℎ 119899 isin N

0 Then

Z [119910] (119911) = (ℎ119911

119911 minus 1)

minus120572

119883 (119911) (42)

where 119883(119911) = Z[119909](119911) and 119909(119899) = 119909(119886 + 119899ℎ)


ℎ119909](119911) = Z[119886(120572)

119904lowast119909][119911] =

Z[119886(120572)119904




ℎ119909)(119899ℎ) =

(0Δ120572


0 or 119909(119905) =




ℎ119910)(119905 + ℎ) =

(119886Δ120572


119886

Proof Let119891(119899) = (0Δ120572



ℎ119910)(0) = 119911(ℎ119911(119911minus


Z[119892] where 119883(119911) = Z[119909](119911) 119909(119899) = 119910(119899ℎ) = 119909(119886 + 119899ℎ) and119892(119899) = (

119886Δ120572

ℎ119909)(119905) 119905 isin (ℎN)




0Δ120572

ℎ119910)(119899ℎ + ℎ) as for the




(0Δ120572

119910) (119905 + ℎ) = 119860119910 (119905) 119905 isin (ℎN)0

119910 (0) = 1199100 1199100isin R119901

(43)




(119860ℎ120572 + 119868120572)Φ(119905) with Φ(0) = 119868



(0Δ120572

ℎ119910) (119905 + ℎ) = 119860119910 (119905) 119905 isin (ℎN)

0

119910 (0) = 1199100 1199100isin R119901

(44)


119910 (119905) = 119864(120572120572)

(119860ℎ120572

119905

ℎ) 1199100 (45)




Φ (119899ℎ) = Zminus1

[(119911

119911 minus 1)120572

(119868 minus1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

] (119899)

(46)




ℎminus119886

Δ120572

ℎlowast)119909)(119905) where 119905 =

119886 + (1 minus 120572)ℎ + 119899ℎ Then

Z [119910] (119911) = ℎminus120572

119911((119911

119911 minus 1)1minus120572

minus 1)119909 (119886) (47)



)] (119911)119909(119886) =


119904=0(minus1)119904

( 120572minus1119904


1)119909(119886)



119864(120572120572)

(119860 119899) = (119864(120572)


) (119899) (48)


Z [119864(120572)


] (119911)

= Z [119864(120572)

(119860 sdot)] (119911)Z [120593120572minus1

] (119911)

=119911

119911 minus 1(119868 minus

1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

(119911

119911 minus 1)120572minus1

= (119911

119911 minus 1)120572

(119868 minus1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

(49)



](119911) =

Z[119864(120572120572)





(119886Δ120572

lowast119909) (119899) = 119860119909 (119886 + 119899) + 119861119906 (119899)

119910 (119899) = 119862119909 (119886 + 119899) + 119863119906 (119899) (50)


(119886Δ120572

119909) (119899) = 119860119909 (119886 + 119899) + 119861119906 (119899)

119910 (119899) = 119862119909 (119886 + 119899) + 119863119906 (119899)(51)




1minus120572

(119911minus

1)119883(119911) = 119860119883(119911) + 119861119880(119911) Then 119883(119911) = (1(119911 minus 1)) sdot



120572119862(119868minus


(120572120572)(119860 119899minus1)119861We


(minus1) = 0




lowast119909eq)(119905) = 119891(119905 119909eq) = 0 (and (

119886Δ120572119894 119909eq)(119905) = 119891(119905 119909eq)


0




0


implies

lim119896rarrinfin

119909 (119886 + 119896ℎ) = 0 (52)




lowast



02

01

0

minus01

minus02

50 100 150 200xvalues

n steps


02

01

0

minus01

minus02

50 100 150 200yvalues

n steps


y

x

02

02

01

01

0

0

minus01

minus01

minus02

minus02




119909(119886 + 119899ℎ) =

lim119899rarrinfin

1205930120572




det(119868 minus1

119911(

119911

119911 minus 1)120572

119860ℎ120572

) = 0 (53)







(1119911)(119911(119911 minus 1))120572

119869ℎ120572

)minus1






119904)


119897 119897 = 1 119904 Let 119896

119894isin N1




119894is 119901119894


(

(

119908minus1 (120582119894) 119908minus2 (120582

119894) 119908minus119903119902 (120582

119894)

0 119908minus1 (120582119894) 119908minus119903119902+1 (120582

119894)

d

0 0 119908minus1 (120582119894)

)

)

(54)


det(119908(119860)) = det(119908(119869)) = prod119898

119894=1119908119896119894(120582

119894) and 119883(119911) is formed by




119899rarrinfin119909(119899) = lim

119911rarr1(119911 minus 1)119883(119911) = 0



are not stable



120572 for all120582 isin Spec(119860)


120572minus2120572

2120572minus2120572 ]


119890= 0 Since




1= minus2120572 minus 2120572119894

1205822







0

y

x

times107

times107

15

10

10

05

minus10

minus10

minus05


8 Conclusions




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119909 (119905) = E(120572120572)

(119860ℎ120572

119905

ℎ) 1199090= 119864(120572120572)

(119860ℎ120572

119899) 1199090

=

infin

sum119896=0

119860119896

ℎ119896120572

120593119896120572

(119899) 1199090

(38)


120572minus1




Z [119872] (119911) = (119911

119911 minus 1)120572

(119868 minus1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

(39)


minus120572





Let 119909 (ℎN)119886


(119886Δ120572






119864(120572120572)


119896=0((120582119896119911119896)(119911(119911 minus 1))119896120572+120572)](119899)





119886Δ120572


119886rarr

R is defined by

(119886Δ120572

ℎ119909) (119905) =

(119905minus119886)ℎ

sum119904=0

119886(120572)

119904119909 (119905 minus 119904ℎ) (40)

where

119886(120572)

119904= (minus1)

119904

(120572

119904)(

1

ℎ120572) with

(120572

119904) =

1 for 119904 = 0


119904 for 119904 isin N

(41)


(119886Δ120572

ℎ119909)(119905) where 119905 isin (ℎN)

119886and 119905 = 119886 + 119899ℎ 119899 isin N

0 Then

Z [119910] (119911) = (ℎ119911

119911 minus 1)

minus120572

119883 (119911) (42)

where 119883(119911) = Z[119909](119911) and 119909(119899) = 119909(119886 + 119899ℎ)


ℎ119909](119911) = Z[119886(120572)

119904lowast119909][119911] =

Z[119886(120572)119904




ℎ119909)(119899ℎ) =

(0Δ120572


0 or 119909(119905) =




ℎ119910)(119905 + ℎ) =

(119886Δ120572


119886

Proof Let119891(119899) = (0Δ120572



ℎ119910)(0) = 119911(ℎ119911(119911minus


Z[119892] where 119883(119911) = Z[119909](119911) 119909(119899) = 119910(119899ℎ) = 119909(119886 + 119899ℎ) and119892(119899) = (

119886Δ120572

ℎ119909)(119905) 119905 isin (ℎN)




0Δ120572

ℎ119910)(119899ℎ + ℎ) as for the




(0Δ120572

119910) (119905 + ℎ) = 119860119910 (119905) 119905 isin (ℎN)0

119910 (0) = 1199100 1199100isin R119901

(43)




(119860ℎ120572 + 119868120572)Φ(119905) with Φ(0) = 119868



(0Δ120572

ℎ119910) (119905 + ℎ) = 119860119910 (119905) 119905 isin (ℎN)

0

119910 (0) = 1199100 1199100isin R119901

(44)


119910 (119905) = 119864(120572120572)

(119860ℎ120572

119905

ℎ) 1199100 (45)




Φ (119899ℎ) = Zminus1

[(119911

119911 minus 1)120572

(119868 minus1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

] (119899)

(46)




ℎminus119886

Δ120572

ℎlowast)119909)(119905) where 119905 =

119886 + (1 minus 120572)ℎ + 119899ℎ Then

Z [119910] (119911) = ℎminus120572

119911((119911

119911 minus 1)1minus120572

minus 1)119909 (119886) (47)



)] (119911)119909(119886) =


119904=0(minus1)119904

( 120572minus1119904


1)119909(119886)



119864(120572120572)

(119860 119899) = (119864(120572)


) (119899) (48)


Z [119864(120572)


] (119911)

= Z [119864(120572)

(119860 sdot)] (119911)Z [120593120572minus1

] (119911)

=119911

119911 minus 1(119868 minus

1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

(119911

119911 minus 1)120572minus1

= (119911

119911 minus 1)120572

(119868 minus1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

(49)



](119911) =

Z[119864(120572120572)





(119886Δ120572

lowast119909) (119899) = 119860119909 (119886 + 119899) + 119861119906 (119899)

119910 (119899) = 119862119909 (119886 + 119899) + 119863119906 (119899) (50)


(119886Δ120572

119909) (119899) = 119860119909 (119886 + 119899) + 119861119906 (119899)

119910 (119899) = 119862119909 (119886 + 119899) + 119863119906 (119899)(51)




1minus120572

(119911minus

1)119883(119911) = 119860119883(119911) + 119861119880(119911) Then 119883(119911) = (1(119911 minus 1)) sdot



120572119862(119868minus


(120572120572)(119860 119899minus1)119861We


(minus1) = 0




lowast119909eq)(119905) = 119891(119905 119909eq) = 0 (and (

119886Δ120572119894 119909eq)(119905) = 119891(119905 119909eq)


0




0


implies

lim119896rarrinfin

119909 (119886 + 119896ℎ) = 0 (52)




lowast



02

01

0

minus01

minus02

50 100 150 200xvalues

n steps


02

01

0

minus01

minus02

50 100 150 200yvalues

n steps


y

x

02

02

01

01

0

0

minus01

minus01

minus02

minus02




119909(119886 + 119899ℎ) =

lim119899rarrinfin

1205930120572




det(119868 minus1

119911(

119911

119911 minus 1)120572

119860ℎ120572

) = 0 (53)







(1119911)(119911(119911 minus 1))120572

119869ℎ120572

)minus1






119904)


119897 119897 = 1 119904 Let 119896

119894isin N1




119894is 119901119894


(

(

119908minus1 (120582119894) 119908minus2 (120582

119894) 119908minus119903119902 (120582

119894)

0 119908minus1 (120582119894) 119908minus119903119902+1 (120582

119894)

d

0 0 119908minus1 (120582119894)

)

)

(54)


det(119908(119860)) = det(119908(119869)) = prod119898

119894=1119908119896119894(120582

119894) and 119883(119911) is formed by




119899rarrinfin119909(119899) = lim

119911rarr1(119911 minus 1)119883(119911) = 0



are not stable



120572 for all120582 isin Spec(119860)


120572minus2120572

2120572minus2120572 ]


119890= 0 Since




1= minus2120572 minus 2120572119894

1205822







0

y

x

times107

times107

15

10

10

05

minus10

minus10

minus05


8 Conclusions




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Φ (119899ℎ) = Zminus1

[(119911

119911 minus 1)120572

(119868 minus1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

] (119899)

(46)




ℎminus119886

Δ120572

ℎlowast)119909)(119905) where 119905 =

119886 + (1 minus 120572)ℎ + 119899ℎ Then

Z [119910] (119911) = ℎminus120572

119911((119911

119911 minus 1)1minus120572

minus 1)119909 (119886) (47)



)] (119911)119909(119886) =


119904=0(minus1)119904

( 120572minus1119904


1)119909(119886)



119864(120572120572)

(119860 119899) = (119864(120572)


) (119899) (48)


Z [119864(120572)


] (119911)

= Z [119864(120572)

(119860 sdot)] (119911)Z [120593120572minus1

] (119911)

=119911

119911 minus 1(119868 minus

1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

(119911

119911 minus 1)120572minus1

= (119911

119911 minus 1)120572

(119868 minus1

119911(

119911

119911 minus 1)120572

119860ℎ120572

)minus1

(49)



](119911) =

Z[119864(120572120572)





(119886Δ120572

lowast119909) (119899) = 119860119909 (119886 + 119899) + 119861119906 (119899)

119910 (119899) = 119862119909 (119886 + 119899) + 119863119906 (119899) (50)


(119886Δ120572

119909) (119899) = 119860119909 (119886 + 119899) + 119861119906 (119899)

119910 (119899) = 119862119909 (119886 + 119899) + 119863119906 (119899)(51)




1minus120572

(119911minus

1)119883(119911) = 119860119883(119911) + 119861119880(119911) Then 119883(119911) = (1(119911 minus 1)) sdot



120572119862(119868minus


(120572120572)(119860 119899minus1)119861We


(minus1) = 0




lowast119909eq)(119905) = 119891(119905 119909eq) = 0 (and (

119886Δ120572119894 119909eq)(119905) = 119891(119905 119909eq)


0




0


implies

lim119896rarrinfin

119909 (119886 + 119896ℎ) = 0 (52)




lowast



02

01

0

minus01

minus02

50 100 150 200xvalues

n steps


02

01

0

minus01

minus02

50 100 150 200yvalues

n steps


y

x

02

02

01

01

0

0

minus01

minus01

minus02

minus02




119909(119886 + 119899ℎ) =

lim119899rarrinfin

1205930120572




det(119868 minus1

119911(

119911

119911 minus 1)120572

119860ℎ120572

) = 0 (53)







(1119911)(119911(119911 minus 1))120572

119869ℎ120572

)minus1






119904)


119897 119897 = 1 119904 Let 119896

119894isin N1




119894is 119901119894


(

(

119908minus1 (120582119894) 119908minus2 (120582

119894) 119908minus119903119902 (120582

119894)

0 119908minus1 (120582119894) 119908minus119903119902+1 (120582

119894)

d

0 0 119908minus1 (120582119894)

)

)

(54)


det(119908(119860)) = det(119908(119869)) = prod119898

119894=1119908119896119894(120582

119894) and 119883(119911) is formed by




119899rarrinfin119909(119899) = lim

119911rarr1(119911 minus 1)119883(119911) = 0



are not stable



120572 for all120582 isin Spec(119860)


120572minus2120572

2120572minus2120572 ]


119890= 0 Since




1= minus2120572 minus 2120572119894

1205822







0

y

x

times107

times107

15

10

10

05

minus10

minus10

minus05


8 Conclusions




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










02

01

0

minus01

minus02

50 100 150 200xvalues

n steps


02

01

0

minus01

minus02

50 100 150 200yvalues

n steps


y

x

02

02

01

01

0

0

minus01

minus01

minus02

minus02




119909(119886 + 119899ℎ) =

lim119899rarrinfin

1205930120572




det(119868 minus1

119911(

119911

119911 minus 1)120572

119860ℎ120572

) = 0 (53)







(1119911)(119911(119911 minus 1))120572

119869ℎ120572

)minus1






119904)


119897 119897 = 1 119904 Let 119896

119894isin N1




119894is 119901119894


(

(

119908minus1 (120582119894) 119908minus2 (120582

119894) 119908minus119903119902 (120582

119894)

0 119908minus1 (120582119894) 119908minus119903119902+1 (120582

119894)

d

0 0 119908minus1 (120582119894)

)

)

(54)


det(119908(119860)) = det(119908(119869)) = prod119898

119894=1119908119896119894(120582

119894) and 119883(119911) is formed by




119899rarrinfin119909(119899) = lim

119911rarr1(119911 minus 1)119883(119911) = 0



are not stable



120572 for all120582 isin Spec(119860)


120572minus2120572

2120572minus2120572 ]


119890= 0 Since




1= minus2120572 minus 2120572119894

1205822







0

y

x

times107

times107

15

10

10

05

minus10

minus10

minus05


8 Conclusions




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119904)


119897 119897 = 1 119904 Let 119896

119894isin N1




119894is 119901119894


(

(

119908minus1 (120582119894) 119908minus2 (120582

119894) 119908minus119903119902 (120582

119894)

0 119908minus1 (120582119894) 119908minus119903119902+1 (120582

119894)

d

0 0 119908minus1 (120582119894)

)

)

(54)


det(119908(119860)) = det(119908(119869)) = prod119898

119894=1119908119896119894(120582

119894) and 119883(119911) is formed by




119899rarrinfin119909(119899) = lim

119911rarr1(119911 minus 1)119883(119911) = 0



are not stable



120572 for all120582 isin Spec(119860)


120572minus2120572

2120572minus2120572 ]


119890= 0 Since




1= minus2120572 minus 2120572119894

1205822







0

y

x

times107

times107

15

10

10

05

minus10

minus10

minus05


8 Conclusions




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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