Fractional calculus operators and their applications involving power functions and summation of series

N O R T H - H O L L A N D

F r a c t i o n a l C a l c u l u s O p e r a t o r s

a n d t h e i r A p p l i c a t i o n s I n v o l v i n g

P o w e r F u n c t i o n s a n d S u m m a t i o n o f S e r i e s

Ming-Po Chen

Institute of Mathematics Academia Sinica Nankang, Taipei 11529 Taiwan, Republic of China

and

H. M. Srivastava

Department of Mathematics and Statistics University of Victoria Victoria, British Columbia VSW 3P4 Canada

ABSTRACT

Many earlier works on the subject of fractional calculus contain interesting accounts of the theory and applications of fractional calculus operators in a number of areas of mathematical analysis (such as ordinary and partial differential equations, integral equations, summation of series, etc.). The main object of this paper is to examine rather systematically (and extensively) some of the most recent contributions on the applications of fractional calculus operators involving power functions and in finding the sums of several interesting families of infinite series. Various other classes of infinite sums found in the mathematical literature by these (or other) means, and their validity or hitherto unnoticed con- nections with some known results, are also considered. (~) Elsevier Science Inc., 1997

1. INTRODUCTION

One of the most frequently encountered tools in the theory of fractional calculus (that is, differentiation and integration of an arbitrary real or complex order) is furnished by the familiar differintegral operator cDz ~,

APPLIED MATHEMATICS AND COMPUTATION 81:287-304 (1997) (~) Elsevier Science Inc., 1997 0096-3003/97/$17.00 655 Avenue of the Americas, New York, NY 10010 SSDI 0096-3003(95)0031D-X

288 M.-P. CHEN AND H. M. SRIVASTAVA

defined by

cD~z{f(z)}

. f (z - - ~ ) - . - l f ( ~ ) d~ (c e R; ~(#) < 0)

d m ~z m cD~z-m{f(z)} (m - 1 < ~(#) < m;

m • N := {1 ,2 ,3 , . . .} ) , (1.1)

provided that the integral exists. For c = 0, the operator D~ given by [cf. Equation (1.1)]

D~z{f(z)} := oD~{f(z)} (# • C) (1.2)

corresponds essentially to the classical Riemann-Liouville fractional derivative (or integral) of order tt (or - # ) . Moreover, when c --* c~, Equation (1.1) may be identified with the definition of the familiar Weyl fractional derivative (or integral) of order # (or - # ) .

In recent years, a great deal of literature has appeared discussing the application of the aforementioned fractional calculus operators in a number of areas of mathematical analysis (cf., e.g., [1-8]; see also a recent paper by Ross et al. [9] dealing with functions with no first-order derivative that might have fractional derivatives of all orders less than 1). In the present paper, we aim at examining rather systematically (and extensively) some of the most recent works on the applications of fractional calculus operators involving power functions and in finding the sums of several interesting families of infinite series. We also consider various other classes of infinite sums (which have been found in the mathematical literature by these (or other) means) and investigate their validity or hitherto unnoticed connec- tions with some known results.

The familiar Leibniz rule for ordinary derivatives admits itself of the following extension in terms of the Riemann-Liouville operator Dz ~ given by (1.2):

D~{f(z)g(z)} = E D~-'~{f(z)}D~{g(z)} n ~ O

( (0 ) t r e e ; := F ( # - ~ + I ) F ( ~ + I ) = # s (#' ~ • e ) ,

which will be required in our present investigation.

(1.3)

Fractional Calculus Operators 289

2. APPLICATION INVOLVING POWER FUNCTIONS

Making use of a certain Eulerian integral representing the Beta function, Vyas and Banerji [10] proved the following fractional integral formula for the power function (cez + ~)x:

cD-J{(az +~)~} = a -~ F(A + 1) (az +~)~+~ r(~ + ~ ¥ 1)

(c = -Z/c~; ~(v) > 0; ~R(A) > -1; a ~t 0; z ~t -~ /a ) . (2.1)

Nishimoto [11, p. 13], on the other hand, similarly proved the equivalent case a -- 1 of this same fractional integral formula (2.1), which essentially is one of the main results in a subsequent work by Tu and Nishimoto [12, p. 37, Theorem 1]. In fact, as already observed by Srivastava and Nishimoto [13], the fractional integral formula (2.1) follows readily from, and is no more general than, the following well-known (rather classical) result (cf., e.g., Erd~lyi et al. [14, p. 185, Equation 13.1(7)]):

r ( ~ + 1) z~+. D ; ~ { z ~ } = r(~ + . + 1) (~(~) > o; ~(~) > -1), (2.2)

which incidentally was used by, among others, S. F. Lacroix in 1819 and P. M. H. Laurent in 1884.

In terms of the Riemann-Liouville operator Dz ~ given by (1.2), Ross [15, p. 87, Theorem 1] made use of the familiar binomial expansion and term- by-term integration in order to prove the fractional integral formula:

n=0 F(-~-- n + 1) F(v + n + 1)

(~(.) > o; ~(A) > -1; I~z/Zl < 1). (2.3)

He also gave a fractional derivative formula by merely replacing v in (2.3) by - v [15, p. 88, Theorem 2].

Since

F(A + 1) : (--1)n(--A)n (A • C; n • No := N tJ {0}), (2.4) F(A - n + 1)

where (A)n denotes the Pochhammer symbol defined by

r(~ + n) ~ I (n = o) (2.5) (A)n . - F(~) -- ~ ~(~ + 1)- . . (~ + n - 1) (n • N),


the fractional integral formula (2.3) can be rewritten at once in the hypergeometric form:

( DZ~{(az + 13)~} - F(v + I-----~ 2FI I , -A; v + I; -

(N(v) > 0; N(A) > -1; I~z//~l < 1), (2.6)

where, as usual, 2F1 denotes the Gauss hypergeometric function. Much more general fractional integral formulas than (2.3) or (2.6) are

already available in the mathematical literature. For example, we have (cf. Erd61yi et al. [14, p. 186, Equation 13.1(9)])

D-[~{zP(z + 7) ~}

-- F (pF(p+I ) I )7~z"+~2FI (p+I ' -A ;+ v u + l ; - ~ )

(~(v) > 0; ~(A) > -1; la rg(z / ' r ) l _< ,r - e (0 < e < .r ) ) , (2 .7)

which, in its special case when p : 0 and 7 : ~ / a , immediately yields Ross's formula (2.6). More generally, in terms of the generalized hypergeometric pFq function, it is known that [14, p. 200, Equation 13.1(95)]

D ~ { z P p F q ( a l , . . . ,ap; i l l , . . . ~q; z)} = F__(p.p+_ 1) . zp+, ' F(p + v + 1)

• p + l F q + l ( p q - 1 , h i , . . . ,ap; p-t- v-b 1,/~l,. . . ,]3q; z)

(~(u) > 0; ~(A) > -1; [z[ < Go when p <_ q; [z[ < 1 when p = q + 1),

(2.8)

which does yield (2.7) in the special case when

p - l = q = O , a l = - A , and z - - ~ - z / 7 .

With a view to presenting a fur ther generalization of the fractional integral formula (2.8), we recall the operator r~'6'' defined by (of. Srivastava et al. [16, p. 413, Equation (1.4)])

/ Z--~--~ f z .,6,., Jo (Z -- ~ ) . - -1 I~, z {f(z)} := C(p)

• 2 F l ( # - b ~ , - ~ l ; #; X - ~ ) f ( ( ) d ¢ , (2.9)

so that, obviously,

p,,--ft,~/ I~,~ {f(z)} = D-[" { f ( z ) } (~R(#) > 0), (2.10)


provided, of course, that the integral in (2.9) exists. For this general fractional integral operator r"6'u it is also known that (cf. [16, p. 415, Equation ~O,z ' (2.3)])

io~,~,u, ~, F ( A + I ) F ( A - 5 + f f + I ) z~_ ~ = 7 + 1)

(N(v) > 0; N(A) > max{0, N(5 - 77)} - 1), (2.11)

which, for 5 = - v , immediately yields the classical result (2.2). By comparing the definitions (1.1), (1.2), and (2.9), it is readily seen

that

I~,~'°{f(z)} = z - " -~D["{ f ( z ) } (!it(#) > 0; 5 • C) (2.12)

and, more interestingly, that

I~, z"'~'~ {f(z) } = ~z_.., (# + 5)n(-y)n_~. z-~,- n D-~,-n{f(z)}, (2.13) n=O

provided that this last series converges. It may be worthwhile to note that, in view of the relationship (2.12), the fractional integral formula (2.11) would reduce to the classical result (2.2) also when ~ = 0.

Making use of the fractional integral formula (2.11), it is not difficult to show that

,z l z p I ~ q ( O Z l , ' ' ' , O ~ P ; ] ~ l , ' ' ' , ] ~ q ; Z)}

r (p + 1)r(p - 5 + ~ + 1) = _ _ Z p - 5

C(p - 5 + 1---)"~(p 7 ; 7 ~ + 1)

• p + 2 F q + 2 z

p - 5 + 1 , p + v + 7 / + 1,f~l,...,~3q;

(~(v) > O; !R(A) > max{-1, ~(5 - ~/ - 1)};

[z[ < oo when p <_ q; [zJ < 1 when p = q + 1) (2.14)

and, more generally, that

oo I;,'~"(zp¢(z)} = ~ r ( p + n + 1 ) r ( p - 5 + ~ + n + 1) - ~ ' n

e ( z ) : = ~ a n z n ; N ( v ) > 0 ; N ( A ) > m a x { - 1 , N ( 5 - r l - 1 ) } , (2.15) n=O

provided that each member of this last result (2.15) exists.


3. THE LEIBNIZ RULE (1.3) AND ITS CONSEQUENCES

The application of the Riemann-Liouville operator D~ in evaluating sums of infinite series is based largely upon the Leibniz rule (1.3). Many of the workers on this subject did indeed revive, as illustrations emphasizing the usefulness of the fractional calculus techniques, various special cases and consequences of the following w e l l - k n o w n (rather classical) result in the theory of the Psi (or Digamma) function ¢(z):

n:ln(A)n - ¢ ( A ) - ¢ ( A - v ) (~R(A-tJ) >0; A # 0 , - 1 , - 2 , . . . ) , (3.1)

where

d r'(z) ¢(z) := {logr(z)} = r - ~ (3.2)

For a detailed historical account of the summation formula (3.1), and of its numerous consequences and generalizations, the reader may be referred to one of the recent works on the subject by Nishimoto and Srivastava [17], who also provided many relevant earl ier references on summation of infinite series by means of fractional calculus. (See also Srivastava [18], A1- Saqabi et al. [19], Aular de Dur£n et al. [20], and Tu and Chyan [21]). These last authors (Tu and Chyan [21]) consider many obvious variations of the summation formula (3.1).

Now we turn to the work of Galu6 et al. [22] who derived the following interesting consequence of the familiar Leibniz rule (1.3):

~--~(_11--1 (#1~ n! D - [ " - k { f (z) } D'~+k {g ( z ) } n = l k = 0

n = l

(3.3)

provided that each side of (3.3) exists. Subsequently, as further applications of (3.3), Galu6 [23] set

(i) f ( z ) = e a~ and g(z) = z b (a # 0); (ii) f ( z ) = z b and g( z ) = e az (a ~t 0);

(iii) f ( z ) = e az and g(z ) = log(bz) (a ~ 0; b ~t 0),

Practional Calculus Operators 293

and deduced from (3.3) the following summation formulas:

.2Fo

1 -m,n-b; -; -

> =I

at (m E W; (3.4)

m

In- ) In n=O

( m + n - 1

n >

(az)”

(b + l)n

. lFl(-m; n + b + 1; --at) = 1 (m E W;

2 (m’~-1)f$-2Fo (-m, n; -; $) n=l

(3.5)

= _ 2 (-mLl r(n) 70” (m E N).

TZ=l

(3.6)

The summation formula (3.5) occurs erroneously in Galue’s paper (23, p. 64, Equation (9)], where the negative sign in the argument of the confluent hypergeometric r FI function is missing. More importantly, the summation formula (3.6) can be shown to follow readily from (3.4). Observe that, since [cf. Definition (2.5)]

(_b) n

= r(n - b, r( -b)

(n E NO),

the summation formula (3.4) can be rewritten at once in its equivalent form:

(

1 -m,n-b; -; -

az

=r(-b){l- 2FO(-m, -b; -; A)} (mEN),

which, when expressed as a (finite) series on the right-hand side, becomes

m+n- 1 r(n- b)

n > (az)” 2Fo ( -m,n-b; -;

1 - az >

= _ 2 C-m), r(n - b) n! (azP

(m E N). n=l

(3.7)

The summation formula (3.6) is an obvious special case of this last result (3.7) when b = 0.


In view of the above observation that the summation formula (3.6) is contained in (3.4), we need only examine the summation formulas (3.4) and (3.5). As a mat ter of fact, a closer examination of these summation formulas would suggest the existence of their unification (and generalization) in the form:

~-'~ (~'t-n-1) (oL1)n'"(O~p)n ~=0 n (/31)n :

• p+iFq ~ ~n .= i ~1 +n,...,~q +n;

(~ ~ c ; ICI < ~ when p _< q - 1; I~l < 1 when p = q), (3.8)

where the constraints upon ~, p, and q may be waived when # = m (m E N). The summation formula (3.4) follows from the general result (3.8) when

1 # = m ( m E N ) , p - l = q = 0 , a l = - b , and ( = az'

while (3.8), in its special case when

# = m ( m c N ) , p = q - l = 0 , ~1 = b + l , and ~=-az, would easily yield the summation formula (3.5).

Our proof of the general summation formula (3.8) is direct (that is, without using fractional calculus techniques). If, for convenience, we denote the first member of (3.8) by A(~), we readily find that

a(¢) = (V)~( - -~ )k(~ l )~+k""" (~p)n+k C+k

,~,k=o (3.9) c~ ( - - , ) k ( O ~ l ) k ' ' ' (Olp)k ~k

= Z (~-5~ ::~-Jqg k~ ~Fl(-k' ~; . - k + 1; 1), k=0

Where the change of the order of summation is justifiable, by absolute convergence of the series involved, under the conditions stated with (3.8). Now, by the Vandermonde summation theorem, we have

(1-k)~ (1 (k=0) ~ F ~ ( - k , ~; ~ - k + 1; 1) - (~ _ k + 1)k 0 (k ~ N), (3.10)

which, when substituted into the last member of (3.9), yields the desired

result: A(~) = 1 (3.11)

under the various constraints stated already with (3.8).


It is not difficult to apply the above proof mutatis mutandis in order to derive the following further generalization of the summation formula (3.8):

f i # + n - 1 --#~kan+k-- A0, (3.12) n ~ 0 n =

provided that the series involved converge absolutely, {A~}n°°= 0 being a suitably bounded sequence of complex numbers.

4. A SYMMETRICAL GENERALIZATION OF THE LEIBNIZ RULE (1.3)

The Leibniz rule (1.3), which was applied by Galu@ et al. [22] in order to derive the summation identity (3.3), suffers from an apparent drawback in the sense that the interchange of the functions f(z) and g(z) on the right- hand side is not obvious. A further (symmetrical) generalization of (1.3), considered by Watanabe [24] and Osler [25], without such a drawback, is given by (cf. e.g., Samko et al. [6, p. 316, Equation (17.12)])

D~z{f(z)g(z)}

= ~-~ ( #a+n) D~-~-n{f(z)}D~+n{g(z)} (#,aeC), (4.1) n ~ - - O 0

which, in the special case when a = 0, yields the Leibniz rule (1.3). By applying the generalized Leibniz ruie (4.1), Aular de Dur~n et al.

[20] derived the following summation identity [20, p. 752, Equation (3.9)]:

( ~ ) ~ C3~n) Dza-~-n{f(z)}D~+~+n{g(z)} n ~ - - O 0 (n#o)

= ,(z.g(z._ ( : )

# D-~-~-~-n-k{f(z)} - ~ a n v + k

(,~#o)

• D~+~+n+k{g(z)} (a, j3,7, # • C), (4.2)

which, in the special case when a = 13 = p = 0, would reduce to the simpler summation identity (3.3).

An obvious speciM case of the summation identity (4.2) when a = 13 = p happens to be the main result of A1-Zamel and Kalla [26, p. 30,


Equation (5)], who also presented several examples illustrating the usefulness of this particular case of (4.2) in deriving various relationships involving infinite sums. The general summation identity (4.2), obtained earlier by Aular de Dur£n et al. [20], is potentially more advantageous in this respect than the special case used by A1-Zamel and Kalla [26].

5. FURTHER APPLICATIONS OF THE GENERALIZED LEIBNIZ RULES

By setting

f ( z ) = 1 and g ( Z ) ~-~ Z N ( N E No)

in the generalized Leibniz rule (1.3), Owa [27] proved the hypergeometric summation formula:

N! 2 F l ( - # , - Y ; 1 - #; 1) = (1 - #)--------~ (Y • N0; # ¢ N). (5.1)

Formula (5.1), as observed also by Owa [27], is a special case of the Gauss summation theorem:

2Fl(a,b;c; 1) = r ( c ) r ( c - a - b) r ( c - )r(c - b)

( ~ ( c - a - b) > 0; c ~ 0 , - 1 , - 2 , . . . ) (5.2)

when a = - # , b = - N , and c = 1 - # (N c N0; /z ¢ N). In fact, the Gauss summation theorem (5.2) would itself emerge from the generalized Leibniz rule (1.3) upon setting

# = - a , f ( z ) = z c - a - l , and g(z) = z -b,

and applying the formula (2.2) on each side. More interestingly, the generalized Leibniz rule (4.1) can be shown to yield the well-known DougaU's formula [2S, p. 7, Equation 1.4(1)]:

~- , r ( a + n ) r (b + n)

r 2 F(c + d - a - b - 1)

s in(ra) sin(rb) r ( c - a ) r ( c - b)r(d - a ) r ( d - b)

(~(c + d - a - b) > 1; a,b ¢ Z := {0, ± 1 , ± 2 , . . . } ) , (5.3)

Fract ional Calculus Operators 297

when we set

# = c - a - l , a = c - l , f ( z ) = Z d - a - l , and g(z) = z c-b-l , (5.4)

and make use of the formula (2.2) on both sides of (4.1). In view of Definition (2.5), a special case of Dougall's formula (5.3)

when d = 1 (or, equivalently, when c = 1) immediately yields the Gauss summation theorem (5.2).

Now we turn to a mild extension of the generalized Leibniz rule (4.1) in the form (cf., e.g., Samko et al. [6, p. 317]):

D f { f ( z ) g ( z ) } = ~ ~ # D2-~*-'~n{f(z)}D~+"'~{g(z)} n=-oo ~r q- a n

(~ , ~ ~ c ; 0 < a < 1), (5.5)

which, for a = 1, reduces at once to (4.1). Indeed, if we choose the pa- rameters # and a (and the functions f and g) just as indicated in (5.4), and apply the formula (2.2) on both sides of (5.5), we shall obtain the summation formula:

E°° F(cF(a ++ an)F(dan)F(b ++ an)an) sin[r(a + an)] sin[~r(b + an)] n ~ - - O O

r 2 F ( c + d - a - b - 1)

a r ( c - a ) r ( c - b ) r ( d - a ) r ( d - b)

( ~ ( c + d - a - b ) > 1 ; a, b C Z ; 0 < a _ < l ) , (5.6)

which, in the special case when a = 1, would readily yield Dougall's formula (5.3).

Since [14, p. 188, Entry (24)]

F(A + 1) ;~ , D~z{z ~ logz} - F(-A---~ + 1) z - " [logz + ¢(A + 1) - ¢(A - # + 1)]

(N(A) > -1 ; N(#) < 0), (5.7)

which yields the following special case when A = 0:

Dz~{l°gz} - r (1 - #) [logz - 3' - ¢(1 - #)] (~(#) < 0), (5.8)

and since

r logz (n = 0) D~{log z} ( - 1 ) n - l ( n - 1)!z -n (n e N), (5.9)


the classical result (3.1) would follow from the generalized Leibniz rule (1.3) when we set

# = - v , f ( z ) = z ~'-v-1, and g ( z ) = l o g z .

More generally, by setting

f ( z ) = z ~-1 and g(z) =logz ,

and applying the formulas (5.7) and (5.S), in conjunction with (2.2), we find from (5.5) that

log z + ¢(A) - ¢(A - #)

o

(~(A) > 0; # , a 6 C; 0 < , < 1), (5.10)

provided further that each member of (5.10) exists. Finally, we rewrite the generalized Leibniz rule (5.5) in the form:

D ~ " { f ( z ) g ( z ) }

= ~ ( - ~ ) D - ~ " - ' ~ { f ( z ) } D ~ { g ( z ) }

+ ~-" ~ a n DZ"-~'-¢"{ f (z)}D~+~n{g(z)} n -~- -- O0

(n#o)

(c~,# 6 C; 0 < ~ < 1), (5.11)

which, in view of (5.5) itself, yields the following (presumably new) summation identity:

n~--O0

(n#O)

,~=-oo k=-oo 7 Ck (n#o)

• DT+~+O~+¢k{g(z)} ( a , ~ , 7 , # 6 C ; 0 < ~ , r / , ¢ < l ) . (5.12)


The summation identity (4.2), given earlier by Aular de Dur£n et al. [20], is an obvious special case of this last result (5.12) when

~=~?--~=1.

6. VALIDITY OF SOME HYPERGEOMETRIC SUMS AND TRANSFORMATIONS

For the Gauss hypergeometric function, the following integral represen- tation is fairly well-known:

1 f0~¢ 2F1(a, b; c; z) - r ( a ) e-tt ~-1 1Fl(b; c; zt)dt

(~(a) > 0; Izl < 1). (6.1)

And, for the confluent hypergeometric function involved in the integrand of (6.1), there exist several multiplication theorems expressing this function in series of similar functions (cf., e.g., Erd~lyi et al. [28, p. 283, Equa- tions 6.14 (1), (2), and (3)]). Making use of these multiplication theorems in (6.1) and applying the Gauss summation theorem (5.2), Samtani and Bhat t [29] proved three transformation formulas for the Gauss hypergeometric function. We choose to recall their main results in the following slightly modified (and, where necessary, corrected) forms (cf. [29, p. 65, Equation (1.1); p. 66, Equations (3.1) and (3.2)]):

r ( e ) r ( c - a - b) 2Fl(a,b; c; z )= ~c(c---a)F~ b) z-a

• 2 F l ( a , a + l - c ; a + b + l - c ; 1 - 1 ) ; (6.2)

2Fl(a,b; c; z) = r ( c ) r ( c - a - b ) z l _ C r ( c - a ) r ( c - b)

. 2 F l ( a + l - c , b + l - c ; a + b + l - c ; l - z ) ; (6.3)

2 F l ( a , b ; c; z ) = r ( c ) r ( ~ - a - b) r ( c - a ) r ( c - b)

• 2Fl(a,b; a + b + l - c ; l - z ) , (6.4)

in each of which the parameter a is assumed to be a negative integer. In view of Euler's transformation [28, p. 64, Equation 2.1.4(23)]:

2Fl(a,b; c; z )= (1 - z )C-a -b2Fl (c -a , c -b ; c; z)

(larg(1 - z)l _ r - e (0 < e < ~)), (6.5)


the transformation (6.3) is an immediate consequence of (6.4). As a matter of fact, each of the transformations (6.2) and (6.4) is itself implied by a known analytic continuation formula for the Gauss hypergeometric function. For example, since

1 - 0 (a = O , - 1 , - 2 , . . . ) , (6.6)

r(a)

the transformations (6.2) and (6.4) would follow immediately from the known results (cf., e.g., Erd@lyi et al. [28, p. 109, Equation 2.10(4); p. 108, Equation 2.10(1)]):

r(c)r(c- a - b )z - a 2Fl(a,b; c; z )= F-~-a)-F(c b)

• 2F1 ( a , a + l - c ; a + b + l - c ; 1 - 1 )

r (c) r (a + b - C)zo_C( 1 z)C-a--b + r(a)r(b)

• 2 F l ( c - a , l - a ; c - a - b + 1 ; 1 - 1 )

Oarg(z) l_<Tr-e ( O < e < r r ) ) (6.7)

and

2Fl(a,b; c; z ) = F(c)F(c-a-b) 2Fl(a,b; a +b+ l - c ; l - z ) r ( c - a)r(c b)

+ r (c) r (a + b - c) (1 - z) c - ° - b r(a)r(b)

• 2F l (c -a , c -b ; c - a - b + l ; l - z )

( l a r g ( 1 - z ) [ < l r - e ( 0 < e < l r ) ) , (6.8)

respectively, whenever the parameter a is restricted to take on negative integer values only. It should be remarked in passing that, since ~R(c - a - b) > 0, the Gauss summation theorem (5.2) can be deduced readily from (6.7) and (6.8) by letting z --* 1.

In the same paper, by setting

1 1 1 1 z=-~ and e = - ~ a + ~ b + ~

in the transformation formulas (6.2), (6.3), and (6.4), but seemingly ignoring the various parametric constraints emerging from their deriva- tions, Samtani and Bhat t [29, Section 4] obtained three strange (and overly


involved) sums for the hypergeometric series

2Fl(a,b; ~a+lb 1 1) 2 +5; '

each of which is significantly different from the well-known sum:

(a,b; 1 + I b + 1 1 ) = F ( X ) F ( ½ a + ½ b + ½ ) 2F1

z a 5 2; V(½a+ ½)V(½b+ ½) ( 1 lb 1 )

~ a + 2 + 5 ¢ 0 ' - 1 ' - 2 ' ' ' " , (6.9)

which is due, in fact, to Kummer [30, p. 134]. The right-hand side of Kummer's formula (6.9) vanishes whenever

or b = - 2 m - 1 ( m E N o ) . a

Furthermore, since

1 -5 1 1) F(½) (6.10) (7) r ( ~ b + _ r(b+ 1) V(½b+ ½)'

by Legendre's duplication formula, and since (by the familiar reflection formula for the F-function)

F( 1 - l a - ½b)F(½ + ½a + ½b) = 1 (6.11) 1 F(½ - ½a + ½b)F(½ + ~ a - ½b)

whenever

a or b=-2m (m6N),

each of the aforementioned hypergeometric sums (derived by Samtani and Bhat t [29, Section 4]) is actually an unnecessarily involved special case of Kummer's formula (6.9) when the parameter b is a negative integer. As a matter of fact, Kummer's formula in its 9eneral form (6.9) would follow from the known analytic continuation formula (6.7) when we set

1 1 1 1 and c - a + b + 5, z = 2 2 2

and apply Kummer's summation theorem [30, p. 134]:

r(a - b + 1)r(~a + 1) 2gl(a,b; a - b +1; - 1 ) = r ( ½ a - b + l ) r ( a + l )

(~(b) < 1; a - b + 1 ~ 0 , - 1 , - 2 , . . . ) (6.12)


in order to sum each of the resulting hypergeometric 2F1(-1) series on the right-hand side of (6.7). The details involved in this derivation of the general result (6.9) may be left as an exercise for the interested reader.

The present investigation was initiated during the second-named author's visits to the Institute of Mathematics (Academia Sinica) at Taipei, the National Chang- Hua University of Education at Chang-Hua, and National Tsing Hua University at Hsin-Chu in July and August 1995. This work was supported, in part, by the National Science Council of the Republic of China under Grant NSC-85-2121-M- 001-013 and, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant 0GP0007353.

REFERENCES

1 V. Kiryakova, Generalized Fractional Calculus and Applications, Pitman Re- search Notes in Mathematics Series 301, Longman Scientific and Technical, Harlow, Essex (John Wiley and Sons, New York), 1994.

2 K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, New York, Chich- ester, Brisbane, Toronto, and Singapore, 1993.

3 K. Nishimoto, Fractional Calculus, Vols. I, II, III, and IV, Descartes Press, Koriyama, 1984, 1987, 1989, and 1991.

4 K. Nishimoto, An Essence of Nishimoto's Fractional Calculus (Calculus in the 21st Century): Integrations and Differentiations of Arbitrary Order, Descartes Press, Koriyama, 1991.

5 K.B. Oldham and J. Spanier, The Fractional Calculus: Theory and Applica- tions of Differentiation and Integration to Arbitrary Order, Academic Press, New York and London, 1974.

6 S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publish- ers, Reading, Tokyo, Paris, Berlin, and Langhorne (Pennsylvania), 1993.

7 H.M. Srivastava and R. G. Bnschman, Theory and Applications of Convolu- tion Integral Equations, Mathematics and Its Applications, Vol. 79, Kluwer Academic Publishers, Dordrecht, Boston, and London, 1992.

8 H.M. Srivastava and S. Owa (Eds.), Univalent Functions, Fractional Calcu- lus, and Their Applications, Halsted Press (Ellis Horwood Limited, Chich- ester), John Wiley and Sons, New York, Chichester, Brisbane, and Toronto, 1989.

9 B. Ross, S. G. Samko, and E. R. Love, Functions That Have no First-Order Derivative Might Have Fractional Derivatives of All Orders Less Than 1, Real Anal. Exchange 20:140-157 (1994/1995).

10 D.N. Vyas and P. K. Banerji, Fractional Integral Formula of the Function (c~z + lJ) a, J. Fractional Calculus 2:83-86 (1992).


11 K. Nishimoto, Power Functions in Fractional Calculus of Nishimoto and That of Lacroix and Riemann-Liouville, J. Fractional Calculus 2:11-25 (1992).

12 S.-T. Tu and K. Nishimoto, On the Fractional Calculus of Functions ( c z - a ) ~ and l o g ( c z - a), Y. Fractional Calculus 5:35-43 (1994).

13 H.M. Srivastava and K. Nishimoto, A Note on a Certain Fractional Integral Formula, J. Fractional Calculus 3:87-89 (1993).

14 A. Erd~lyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of Inte- gral Transforms, Vol. II, McGraw-Hill Book Company, New York, Toronto, and London, 1954.

15 B. Ross, A Formula for the Fractional Integration and Differentiation of (a + bx) c, J. Fractional Calculus 5:87-89 (1994).

16 H . M . Srivastava, M. Saigo, and S. Owa, A Class of Distortion Theorems Involving Certain Operators of Fractional Calculus, J. Math. Anal. Appl. 131:412-420 (1988).

17 K. Nishimoto and H. M. Srivastava, Certain Classes of Infinite Series Summable by Means of Fractional Calculus, J. College Engrg. Nihon Univ. Set. B 30:97-106 (1989).

18 H.M. Srivastava, A Simple Algorithm for the Evaluation of a Class of Gen- eralized Hypergeometric Series, Stud. Appl. Math. 86:79-86 (1992).

19 B. N. A1-Saqabi, S. L. Kalla, and H. M. Srivastava, A Certain Family of Infinite Series Associated with Digamma Functions, J. Math. Anal. Appl. 159:361-372 (1991).

20 J. Aular de Dur£n, S. L. Kalla, and H. M. Srivastava, Fractional Calculus and the Sums of Certain Families of Infinite Series, Y. Math. Anal. Appl. 190:738-754 (1995).

21 S.-T. Tu and D.-K. Chyan, A Certain Family of Infinite Series, Differinte- grable Functions and Psi Functions, J. Fractional Calculus 7:41-46 (1995); see also Preceedings of the First International Workshop on Transform Meth- ods and Special Functions (Bankya, Bulgaria; August 12-17, 1994) (P. Rusev, I. Dimovski, and V. Kiryakova, Eds.), SCT Publishing Company, Singapore, 1995, pp. 310-316.

22 L. Galu@, S. L. Kalla, and K. Nishimoto, Application of Fractional Calculus to Infinite Sums, J. Fractional Calculus 1:17-21 (1992).

23 L. Galu@, Application of Fractional Calculus to Infinite Sums (II), J. Frac- tional Calculus 7:61-67 (1995).

24 Y. Watanabe, Notes on the Generalized Derivative of Riemann-LiouviUe and Its Applications to Leibniz's Formula. I and II, TShoku Math. J. 34:8-27 and 28-41 (1931).

25 T . J . Osler, Leibniz Rule for Fractional Derivatives Generalized and an Ap- plication to Infinite Series, SIAM J. Appl. Math. 18:658-674 (1970).

26 A. A1-Zamel and S. Kalla, An Application of Generalized Leibniz Rule to Infinite Sums, J. Fractional Calculus 7:29-33 (1995).

27 S. Owa, An Application of the Fractional Derivative to Hypergeometric Se- ries (in Japanese), Sugaku 38:360-362 (1986).


28 A. Erd@lyi, W. Magnus, F. Oberhettinger, and F.G. Tricomi, Higher Tran- scendental Functions, Vol. I, McGraw-Hill Book Company, New York, Toronto, and London, 1953.

29 R . K . Samtani and R. C. Bhatt, A Useful Hypergeometric Transformation, Gan. ita Sandesh 8:65-67 (1994).

30 E . E . Kummer, Uber die hypergeometrische Reihe

1 + c~ " ~ x + a ( a + 1)~(f~ + 1)x2 a(a + 1)(a + 2)~(f~ + 1)(/3 + 2)x a 1- 172 . ' y (~ + 1) + 1 • 2 . 3 . ~/(~ + 1)(3, + 2) + ' ' "

J. Reine Angew. Math. 15:39-83 and 127-172 (1836).

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