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Mathematics Today Vol.27(Dec-2011) 24-39 ISSN 0976-3228
On New Prajapati-Shukla Functions And Polynomials
J. C. Prajapati1
and A. K. Shukla2
1Department of Mathematical Sciences,
Faculty of Technology and Engineering,
Charotar University of Science and Technology,
Changa, Anand-388 421, India.
2Department of Applied Mathematics and Humanities,
S.V.National Institute of Technology,
Surat-395 007, India
E-mails: jyotindra18@rediffmail.com
ajayshukla2@rediffmail.com
ABSTRACT
The principal aim of the paper is to introduce the various generalizations of
Shukla-Prajapati functions and polynomials. For these new functions and
polynomials their various properties including usual differentiation and
integration, Integral transforms, Generalised hypergeometric series form, Mellin
– Barnes integral representation, Recurrence relations, Integral representation,
Decomposition, Fractional calculus operators properties, generating relations,
bilateral generating relation and finite summation formulae of new class of
polynomials also established.
Key Words: Mittag–Leffler function, Integral Transforms, Fractional integral and differential
operators.
AMS classification(2000): 33E12, 33C20, 44A20, 30E20.
J. Prajapati & A. Shukla - On New Prajapati-Shukla Functions And Polynomials 25
1. INTRODUCTION AND PRELIMINARIES
Recently, Shukla and Prajapati (2007), investigated and studied the function
)(,
, zE q which is defined for ,, C; 0Re,0Re,0Re and )1,0(q N
as:
!)(
0
,
,n
z
nzE
n
n
qnq
, (1.1)
where )(
)()(
qnqn denotes the generalized Pochhammer symbol Rainville(1960)
which in particular reduces to
q
r n
qn
q
rq
1
1 if q N.
In continuation of the study, the generalization of (1) can be written as )(,
, zE which
is defined for 0Re,0Re,0Re and 0 as
!)(
0
,
,n
z
nzE
n
n
n
. (1.2)
This is a generalization of the exponential function exp(z), the confluent hypergeometric
function z;, Rainville (1960), the Mittag – Leffler function Mittag-Leffler (1903),
the Wiman’s function Wiman (1905) and the function )(, zE defined by Prabhakar (1971)
as well as equation (1.1).
Gorenflo et al. (1998), Gorenflo and Mainardi (2000), Kilbas et al. (1996, 2004), Saigo and
Kilbas (1998), Srivastava and Tomovski (2009), Tomovski et al (in press) and many other
researchers also studied the various properties of Mittag-Leffler functions and its
generalizations with their applications. The applications of Integral Transforms discussed by
Sneddon (1979)
The function )(,
, zE is an entire function of order 1
1Re
if 1Re , and
absolutely convergent in 1, RRz if 1Re .
The truncated power series of the function )(,
, zE can be defined as,
N
m
N
m
n
j n
jmn
jmn
m
nm
mnN
nm
z
mm
zzE
0
1
0
1
1
,,
,1
1)(
)(
!)(
)()(
(1.3)
and a special case for the study of )(,, zEq for
n
1 as:
0 0
1
1
,
,1
1)(
)(
!)(
)()(
m m
n
j n
jmn
jmn
m
nm
m
nm
z
mm
zzE
(1.4)
where 2n , 1N ; 0Re , 0Re and 0 .
26 Mathematics Today Vol.27(Dec-2011) 24-39
Authors investigated the operator fE aw
,
;,, as,
)()(])([)()()( ,
,
1,
;,, axdttftxwEtxxfE
x
a
aw
. (1.5)
where w C ; 0Re,0Re,0Re ; 0 .
If 1 , then (1.5) reduces to the result Prabhakar (1971).
Shukla and Prajapati (2008 A) introduced a class of polynomials which are connected by
Mittag-Leffler function )(xE , in continuation of the study of class of polynomials, Authors
introduced a general class of polynomial defined as,
xpExxpEn
xskaxA k
n
k
na
n
,
,
,
,
,,,
!,,; (1.6)
where ,,, are real or complex numbers; ska ,, are constants and generalized Mittag-
Leffler function defined as (1.2).
The proofs of all results established in this paper are parallel to Shukla and Prajapati (2007 A,
2007 B, 2007 C, 2008 A, 2008 B, 2009 A, 2009 B, 2010).
2. BASIC PROPORTIES OF THE FUNCTION )(,
, zE
As a consequence of the definitions (1.2) the following results hold:
THEOREM (2.1). If ,, C; 0Re,0Re,0Re and 0 then
)()()( ,
1,
,
1,
,
1, zEdz
dzzEzE
(2.1)
0
1,1
,
,
,!
)()(n
n
n
nn
zzzEzE
(2.2)
in particular,
)()()( ,1
,, zzEzEzE
. (2.3)
THEOREM (2.2). If w,,, C; 0Re,0Re,0Re and 0 then
for m N,
zEzEdz
d m
mm
m
,
,
,
,
(2.4)
zEzwzEzdz
d
m
mm
,
,
1,
,
1
, 0Re m (2.5)
J. Prajapati & A. Shukla - On New Prajapati-Shukla Functions And Polynomials 27
in particular,
zEzwzEzdz
dm
mm
,1
,1 (2.6)
and
wzmz
mwzz
m
dz
d m;,;,
11
. (2.7)
THEOREM (2.3). If r
p with rp, N relatively prime; , C and 0 then
)(,
,
r
p
r
pp
p
zEdz
d
!
1
1
1
1 n
pr
n
z
pr
np
r
np
r
npn
n
, (2.8)
in particular,
r
n
zr
n
zezE
r
n
r
r 1
,1
)(
1
1
1
1
, (r = 2, 3 . . .) . (2.9)
3. GENERALIZED HYPERGEOMETRIC FUNCTION REPRESENTATION OF
)(,, zEq
Using (1.2), taking k N and N then we have
;;
;;1,,
z
FzEk
. (3.1)
Convergence criteria for generalized hypergeometric function kF :
(i) If k , the function kF converges for all finite z.
(ii) If 1 k , the function kF converges for 1z and diverges for 1z .
(iii) If 1 k , the function kF divergent for 0z .
(iv) If 1 k , the function kF is absolutely convergent on the circle 1z if
.011
Re1 1
k
j i
i
k
j
28 Mathematics Today Vol.27(Dec-2011) 24-39
where ; is a -tuples
1,...,
1,
;
; is a -tuples
1,...,
1,
.
4. MELLIN-BARNES INTEGRAL REPRESENTATION OF )(,
, zE
THEOREM (4.1). Let ,;R C ( 0 ) and N. The function )(,
, zE is
represented by the Mellin – Barnes integral as:
dszs
qss
izE s
L
)(
)(
)()(
)(2
1)(,
,
, (4.1)
where zarg ; the contour of integration beginning at i and ending at , i and
indented to separate the poles of the integrand at ns for all n N 0 (to the left) from
those at
q
ns
for all n N 0 (to the right).
5. INTEGRAL TRANSFORMS OF )(,
, zE
In this section, some useful integral transforms like Euler transforms, Laplace transforms,
Mellin transforms and Whittaker transforms also discussed.
THEOREM (5.1). Euler (Beta) transforms
1
0;,,,
;,,,)1(
22
,
,
11
ba
xabdzxzEzz ba
, (5.1)
where 0Re,0Re,0Re,0Re,0Re,0Re ba and 0 .
THEOREM (5.2). Laplace transforms.
0 ;,
;,,,
12,
,
1
s
xaasdzxzEez za s
, (5.2)
where 0Re,0Re,0Re,0Re,0Re a , 0 and .1s
x
J. Prajapati & A. Shukla - On New Prajapati-Shukla Functions And Polynomials 29
THEOREM (5.3). Mellin transforms.
0
,
,
1 )( dtwtEt s
sw
sss
, (5.3)
where 0Re,0Re,0Re,0Re s and 0 .
To obtain Whittaker transform, we use the following integral,
01
2
1
2
1
,12
v
vv
dttWte vt
, where 2
1Re v . (5.4)
THEOREM (5.4). Whittaker Transforms.
0
,
,,21
1dtwtEptWet
pt
;,1,,
;,2
1,,
23
p
wqp
, (5.5)
where 0Re,0Re,0Re,0Re,0Re and 0 .
6. RECURRENCE RELATIONS
THEOREM (6.1). For any Re 0 p , Re 0 s and Re 0 , 0 we get
zEzE spsp
,
2,
,
1,
zEzspp
zEzpzEss
sp
spsp
,
3,
,
3,
22,
3,
12
2
(6.1)
where )]([)( ,
,
,
, zEdz
dzE qq
and )]([)( ,
,2
2,
, zEdz
dzE qq
.
THEOREM (6.2). For k and m N
)()()2()()]1(2[)()( ,
2,
,
3,
,
3,
,
3,
22,
1, zEzEmmzEmkkzzEzkzE mkmkmkmkmk
.
(6.2)
7. INTEGRAL REPRESENTATIONS
THEOREM (7.1). For any Re 0 p , Re 0 s and Re 0 , 0 and setting
kp , ms , where k and m N then
1
0
,
, )( dttEt k
mk
m = )1()1( ,
2,
,
1,
mkmk EE (7.1)
30 Mathematics Today Vol.27(Dec-2011) 24-39
THEOREM (7.2). If 0Re,0Re,0Re,0Re and 0 then
dttEtzt
z
0
,
,
11 )()()(
1
= )(,
,
1
zEz
. (7.2)
Special cases of Theorem (7.2): For 0Re , from (7.2), the particular cases listed as below:
dtetz
z
t
0
1)()(
1
= )(1,1
1,1 zEz
, (7.3)
dtttz
z
0
1 )cosh()()(
1
= )( 21,1
1,2 zEz
, (7.4)
dtt
ttz
z
0
1)sinh(
)()(
1
= )( 21,1
2,2 zEz
, (7.5)
dtzerfztz c
z
)(exp)()(
1
0
21
= )(1,1
1,2
1 zEz
. (7.6)
THEOREM (7.3). If 0Re,0Re,0Re and 0 then
dttEtzt
z
0
2,
,2
11 )()()(
1
= )]()([ 2,
,2
,
,
1
zEzEz . (7.7)
THEOREM (7.4). If 0Re,0Re,0Re and 0 then
0
1,
,4 )(
2
dxxxEe t
x
=
2,
2
1,
2
2
tEt . (7.8)
THEOREM (7.5). If 0Re,0Re,0Re and 0 then
)()()()( ,
,
,
1,
2,
,
1
axEaxExaxExdx
d
. (7.9)
THEOREM (7.6). If 0Re,0Re,0Re and 0 then
)(1 ,
,
1
,
1
ztEts
zsL
. (7.10)
J. Prajapati & A. Shukla - On New Prajapati-Shukla Functions And Polynomials 31
THEOREM (7.7). If 0Re,0Re,0Re , 0 and 0 then
0
1
0
,
,!)(
)(exp)(
n k
n
n
n
k
k
dtnn
tt
z
tzkzE
. (7.11)
dtztEtzE qq )()1()(
1)( ,
,
1
0
1
1
,
,
. (7.12)
dttzEttzE qq
)1()1(
)(
1)( ,
,
1
0
11,
,
. (7.13)
THEOREM (7.8). If 0Re,0Re,0Re,0Re,0Re,0Re , C
and
0 then
)()()1()(
1 ,
,
1
0
,
,
11 zEduzuEuu
. (7.14)
)()()()()()(
1 ,
,
1,
,
11 txEtxdstsEtssx
x
t
. (7.15)
If 1 then
wxExdtwtEtxwEtxt v
v
x
v
v 1,
,
1
0
1,
,
1,
,
11 )()()(
. (7.16)
wzEzdtwtEt
z
,
1,
0
,
,
1
. (7.17)
in particular,
wzEzdtwtEt
z
1,
0
,
1
(7.18)
and );1,();,(0
1 wzz
dtwtt
z
. (7.19)
32 Mathematics Today Vol.27(Dec-2011) 24-39
8. DECOMPOSITION OF MITTAG-LEFFLER FUNCTION
THEOREM (8.1). Integral Representations of the function )(,
,1 zEn
)(,
,1 zEn
= )(,
,1
nzE +
duuuzEn kn
n
k
z
nn
nk
11
1 0
,
,1 )(1
1
, (8.1)
where 2n ; 0Re , 0Re and 0 .
THEOREM (8.2). If 2n , 0Re , 0Re and 0 then
n
n
zEzdz
d1
,
,1
1
1
1
)(1
1
)(1
!)(
)(
!)(
)( n
k nm
n
m
nmn
k nk
n
k
kn
nm
z
kn
zz
(8.2)
THEOREM (8.3). )(,
,1 zEn
n
nk
kn
k
zEz
Re1
,
,1
1
0
, (8.3)
where 2n ; 0Re , 0Re and 0 .
Remark of Theorem 8.3: It is easy to verify that,
zEzE nN
n
nN
n
,,
,1
,,
,1 )(
zE
n
,
,1
n
nk
kn
k
zEz
,
,1
1
0 1
. (8.4)
THEOREM (8.4). If 2n ; 0Re , 0Re and 0 then
)()( .,
,1
,
,1 zEzE nN
nn
T (8.5)
where
nn
k nnNk
nNk
N
Nn
NzE
z
NNN
zT Re
1)(
)(
!)1()1(
)(,
,1
1
1
1
)1(
. (8.6)
9. FRACTIONAL INTEGRAL AND DIFFERENTIAL OPERATORS
ASSOCIATEDWITH THE FUNCTION )(,
, zE
The following well-known facts are prepared for studying properties of the Riemann-
Liouville fractional integrals and differential operators associated with the function )(,
, zE q
and also the properties of operator fE aw
,
;,, .
J. Prajapati & A. Shukla - On New Prajapati-Shukla Functions And Polynomials 33
• ),( baL Space of Lebesgue measurable real or complex valued functions Kilbas et
al.(2004):
),( baL Consists of Lebesgue measurable real or complex valued functions )(xf on ],[ ba
i.e. ),( baL =
b
a
dttfff )(:1
. (9.1)
Kilbas et al. (2004) studied the several properties of fractional integral and differential
integral
operators.
• Confluent hypergeometric functions Rainville (1960):
This is also known as the Pochhammer – Barnes confluent hypergeometric function defined
as,
0
11!)(
)();;();,(
n n
n
n
nb
zazbaFzba , (9.2)
where 0b or a negative integer is convergent for all finite z.
• Gauss multiplication theorem Rainville (1960):
If m is a positive integer and z C then,
)()2(1
2
1
2
1
1
mzmm
kz
mzmm
k
. (9.3)
• Riemann-Liouville fractional integrals of order Khan and Abukhammash GS (2003)
Let ),()( baLxf , C )0)(Re( then
)(xfI xa
= )(xfIa
= dt
tx
tfx
a
1)(
)(
)(
1 ( ax ) (9.4)
is called R-L left-sided fractional integral of order .
Let ),()( baLxf , C ( 0)Re( ) then
)(xfIbx
= )(xfIb
= dt
xt
tfb
x
1)(
)(
)(
1 ( bx ) (9.5)
is called R-L right-sided fractional integral of order .
34 Mathematics Today Vol.27(Dec-2011) 24-39
Theorem (9.1). Let ),0[ Ra and let w C; 0)Re(),Re(),Re(),Re( ,
0 for ax , then
)(})({)[( ,
,
1 xatwEatIa
= ])([)( ,
,
1
axwEax
(9.6)
and
)(})({)[( ,
,
1 xatwEatDa
= ])([)( ,
,
1
axwEax
. (9.7)
Theorem (9.2). Let ,, C and 0 then
)]([ ,1
1,0
xEI x = )(,1
1,
xEx (9.8)
in particular,
)()( 1,10 xExeI x
x
. (9.9)
Theorem (9.3). Let ),0[ Ra and let w C; 0)Re(),Re(),Re(),Re( ,
0 for ax , then
)()( 1,
;,, xatE q
aw
=
)()()( ,
,
1 axwEax q
. (9.10)
Theorem (9.4). Let ),0[ Ra and let w C; 0)Re(),Re(),Re(),Re( ,
0 for ab then the operator q
awE ,
;,,
is bounded on ),( baL and
11
,
;,, fBfE q
aw
(9.11)
where !
)(
)]Re()[Re()(
)()(
)Re(
0
)Re(
k
abw
kkabB
k
k
qk
(9.12)
the relation fEfEI q
aw
q
awa
,
;,,
,
;,,
(9.13)
hold for any summable function ),( baLf .
We can express the function )(,
, zEm
, m N as:
1
0 02
1
2
1
,
,
!
)(1)2()(
m
k nnm
n
n
n
m
mm
z
nm
k
m
km
zE
. (9.14)
J. Prajapati & A. Shukla - On New Prajapati-Shukla Functions And Polynomials 35
On substituting 1 in (9.14) and then the result Kilbas et al. (2004) becomes a special case
of (9.14) as:
1
02
1
2
1
,
1,
, ;,1)2(
)()(m
km
m
mmm
z
m
k
m
km
zEzE
. (9.15)
10. FRACTIONAL OPERATORS AND FUNCTION )(,
, zE
Consider the function
0
2)!(
)()()(
n
n
n
n
cttf
, where 0)Re( , 0
and c is an arbitrary constant then the fractional integral operator of order can be written as,
)()( ,
1,1 ctEttfI
. (10.1)
We denote the function (10.1) as ),,,( qcEt , i.e. )(),,,( ,
1,1 ctEtcEt
. (10.2)
The fractional differential operator of order can be written as
02)!(
)()()(
n
n
nkn
n
ctIDtfD
)(,
1,1 ctEt
. (10.3)
We denote the function (10.3) as ),,,( qcEt , i.e. )(),,,( ,
1,1 ctEtcEt
(10.4)
THEOREM (10.1). If 0)Re( , 0 , c is an arbitrary constant and fractional integral
operator of order then
),,,(),,,( cEcEI tt . (10.5)
),,,(),,,( cEcED tt . (10.6)
The Laplace transforms of ),,,( cEt is given as
,
11
1)},,,({
s
c
scEL t , (10.7)
In the light of Theorem (10.1), we can prove following Theorem (10.2).
THEOREM (10.2). If 0)Re( , 0 , c is an arbitrary constant and fractional integral
operator of order then
),,,(),,,( cEqcEI tt . (10.8)
36 Mathematics Today Vol.27(Dec-2011) 24-39
),,,(),,,( cEqcED tt . (10.9)
,
11
1)},,,({
s
c
scEL t . (10.10)
11. GENERATING RELATIONS AND FINITE SUMMATION FORMULAE OF (6)
A considerably large number of special functions (including all of the classical orthogonal
polynomials) are known to possess generating relations.
We used operational technique by employing Mittal (1977), Patil and Thakare (1975) as a
differential operator, where )( xDsxa and )1(1 xDxa , dx
dD , for obtaining
following generating relations and finite summation formulae of (1.6).
0
),,,( ),,;(n
n
n tskaxA = }])1({[)}({)1(
1,
,
,,
akk
as
atxpExpEat
. (11.1)
!),,;(
0
),1,1,1(
0 m
tskaxA
a
u m
m
km
n
n
n
0
1
),1,1,1( ,,;n
nk
k
k
n
t
a
uska
txAe
. (11.2)
0
),,,( ),,;(n
nan
n tskaxA = }])1({[)}({)1(
1,
,
,,
1 akk
as
atxpExpEat
. (11.3)
0
),,,(
)( ),,;(m
m
nm tskaxAm
mn
=
},,;)1(
}])1({[
)}({)1(
1{
),,,(
1
,
,
,
,skaatxA
atxpE
xpEat a
n
a
k
kasn
. (11.4)
0
),,,(
)( ),,;(n
nan
nm tskaxAm
mn
=
},,;)1(
}])1({[
)}({)1(
1{
),,,(
1
,
,
,
,1skaatxA
atxpE
xpEat a
m
a
k
kas
. (11.5)
J. Prajapati & A. Shukla - On New Prajapati-Shukla Functions And Polynomials 37
Using (11.1) to (11.5), we obtained following generating relations
0
),,,(
)( ),,;(m
m
nm
n zskaxAm =
}])1({[
)}({)1(
1
,
,
,
,
a
k
k
atxpE
xpEaz a
s
m
am
n
m az
zskaazxAmnSm
1
],,;)1([),(!
1
),,,(
0
(11.6)
)0;;(1
0
aazNn
0
),,,(
)( ),,;(m
mam
nm
n zskaxAm =
}])1({[
)}({)1(
1
,
,
,
,1
a
k
k
atxpE
xpEaz a
s
m
aam
m
n
m az
zskaazxAmnSm
1
],,;)1([),(!
1
),,,(
0
. (11.7)
)0;;(1
0
aazNn
Two finite summation formulae for (1.6) also obtained as
),,;(!
1),,;( )0,,,(
)(
0
),,,( skaxAa
am
skaxA mn
m
n
n
m
n
. (11.8)
),,;(!
1),,;( ),,,(
)(
0
),,,( skaxAa
am
skaxA mn
m
n
n
m
n
. (11.9)
we get following bilateral generating relation for ),,;(),,,( skaxAqn
,
0
,
),,,(
)( )(),,;(m
mb
mm tyRskaxA
=
],)1([
}])1({[
)}({)1(
1
,1
,
,
,
, ba
b
a
k
kasn
ytatx
atxpE
xpEat
, (11.10)
where
0
),,,(
)(,, ),,;(],[m
m
bmmb tskaxAtx , 0, m
38 Mathematics Today Vol.27(Dec-2011) 24-39
and )(, yRb
m is a polynomial of degree
b
m in y, which is defined as
)(, yRb
m =
bm
k
k
k ybk
m/
0
,
; b is a positive integer, is an arbitrary complex number.
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