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Integral Transforms and Special Functions

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The convolution method for the development of new leibniz rules involving fractional derivatives and of their integral analogues

H.M. Srivastava a; S.B. Yakubovich b; Yu.F. Luchko b a Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, Canada b Department of Mathematics and Mechanics, Byelorussian State University, Minsk 50, Byelarussia

To cite this Article Srivastava, H.M., Yakubovich, S.B. and Luchko, Yu.F.(1993) 'The convolution method for the

development of new leibniz rules involving fractional derivatives and of their integral analogues', Integral Transforms and Special Functions, 1: 2, 119 — 134 To link to this Article: DOI: 10.1080/10652469308819014 URL: http://dx.doi.org/10.1080/10652469308819014

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Integral Transforms and Speczal Functzons, 1993, Vol. 1, No. 2 , pp.119-134

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THE CONVOLUTION METHOD FOR THE DEVELOPMENT O F NEW LEIBNIZ RULES INVOLVING FRACTIONAL DERIVATIVES AND O F THEIR INTEGRAL ANALOGUES H.M. SRIVASTAVA~,S.B. YAKUBOVICH~, and Yu.F. LUCHK02 'Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3P4, Canada Departmen t of Mathematics and Mechanics, Byelorussian State University, 220050 Minsk 50, Byelarussia

Downloaded At: 20:58 5 January 2010

(Received September 2, 1992) Leibniz rules for the operat,or of fractional derivatives and their integral analogues were considered, among other workers on the subject of fractional calculus, by T.J. Osler and Y. Watanabe. This operator of fractional derivatives is known to belong to a class of integral transforms associated with the Mellin convolution. Since Mejer's G-function transformation happens to be one of the most general integral transforms of this class, it would seem natural to obtain some new Leibniz rules and their integral analogues for operators involving the G-transforms. The object of the present paper is to summarize the development and applications of a general method for constructing such Leibniz rules and their integral analogues based upon the notion of the G-convolution and upon various representations of the kernels of G-convolutions.

KEY WORDS: Leibniz rules, G-transform, G-convolution, fractional derivatives, Meijer's Gfunction

1. INTRODUCTION

The usefulness in mathematical analysis of the familiar Leibniz rule for the operator of (ordinary) derivatives cannot be overemphasized. Some of the important generalizations of this rule, involving the operator D& of fractional derivatives (cf., e.g., [I]; see also [9]), include

120

H.M. SRIVASTAVA, S.B. YAKUBOVICH, and Yu.F. LUCHKO

or, in the symmetrical form,


which admits itself of an integral analogue in the form:

where a and p are arbitrary real or complex numbers. Several workers of the subject of fractional calculus (e.g., Watanabe [12] and Osler [ 5 ] ,[6]; see also Samko et al. [9]) have considered such Leibniz rules involving fractional derivatives as (1)) (2), and (3). In the literature on integral transformations (cf., e.g., [I]), it is already known that the fractional derivative operator D&, involved in the Leibniz rules ( I ) , (2), and (3) above, belongs t o the class of integral transforms associated with the Mellin convolution. Since Mejer's G-function transformation happens t o be one of the most general integral transforms of this class (see [I]), we find it t o be natural t o look for some new Leibniz rules and their integral analogues for operators involving the G-transforms. In the present paper we aim a t summarizing the development and applications of a general method for the construction of such Leibniz rules and their integral analogues. Our method is based upon the notion of the Gconvolution and upon various representations of the kernels of G-convolutions.

2. THE N O T I O N O F THE G - C O N V O L U T I O N

We begin by recalling the following general notions:

Definition 1 (Tuan et al. [ I l l ) . Let c, y E R , and 2sgn(c)+sgn(y) _> 0, where sgn is the signum function. Denote by M;+(L) the space offunctions f (x), x E (0, a), representable in the form:

$1.

where f * ( s ) l s ( ~ ~ ~ ~El L ' ~( a( )~, a) I= {s : Re(s) = T h e space M,+(L) is a Banach space with the norm given by

THE CONVOLUTION METHOD INVOLVING FRACTIONAL DERIVATIVES

121

Obviously, the set M,+(L) of spaces is well-ordered, that is,

for 2sgn(cl - c)

+ sgn(y1 - 7) 2 0.

Definition 2 (Marichev [4] and Srivastava et al. [lo]). Meijer's G-function is defined, as usual, by the following contour integral:

where

The contour L in the definition (4) may be of one of three kinds: L-,, L+, or L;,. The contour L-, (or L+,) is the left (or right) loop which is located in some horizontal strip of the complex s-plane; it begins at the point s = -m iX (or s = cc + iX), leaves all poles at s = -,Bj - I ( j = 1 , . . . , m ; I = 0 , 1 , 2 , . . . ) of the integrand @(s) to the left, and all poles at s = 1 - aj 1 ( j = I , . . . , n ; I = 0 , 1 , 2 , . . . ) to the right of the contour, and then ends at the point s = -cc+ib (or s = cc iS), where X < 6. The contour Li, begins a t the point s = y - icc and ends at the point s = y +ice, separating the aforementioned poles in the same way as L-, or L+, does. The integral (4) converges under any of the following four conditions: (a) L = ti,: c* > 0, 1 arg(z)l < c*T ; (b) L = Ci,: C* 2 0, larg(z)( = C*T, (q - p)y < - Re(p); p < q, 0 < lzl < a; or p = q, 0 < 1t.l < 1; or p = q, c* 2 0, (c) L = L-,: J z J= 1, Re(,u) < 0; < cc; or p = q, (zl > 1; or p = q, c* 2 0, It1 = 1, (d) L = L+,: p > q, 0 < Re(p) < 0. Here, and in what follows, we have


+

+

+

c* = m

+ n - #1P + y),

y = lim {Re(s)) a ---r 00

and

( s E La,),

H.M. SRIVASTAVA, S.B. Y A K U B O V I C H , and Yu.F. LUCHKO

122

Remark 1. If, for the function G:bn(x) hold true:

defined by (4), the following conditions

2sgn(c*)+sgn(y*-1)>O; Re(crj) then, for any

E

Re(Pj)>-1

< -2

1 2

( j = l , . . . , m);

( j = l l. . . , n ) ,

> 0 [2sgn(c*) + sgn(y* - 1 - E )

> 01,


Definition 3 (Tuan et al. [ll]). The G-transform of a function f (x) is defined by

where @(s) is given by (5)) f * ( s ) = M { f(x); s ) denotes the Mellin transform of and the complex parameters ol,, . . , a p the function f ( x ) , o = {s : Re(s) = and P I , . . . , Pq are constrained by the following inequalities:

k),

Re(aj)

1

< -2

(j = 1

n),

Re(aj)

1

> --2

(j =n

+ 1 , . . . ,p).

The G-transform defined by (7) is known to exists for functions belonging to the space M ; + ( L ) , if 2sgn(c c*) sgn(y y*) 0, where c* and y* are given by (6), provided further that it can be represented in the form:

+ +

under the additional condition: 4 sgn(c*)

+

>

+ 2 sgn(y*) + sgn Ip - ql > 0.

Remark 2. The G-transform (7) includes many interesting particular cases (see, for details, [I]); we recall here some of these special cases which we shall use in our present investigation:

T H E CONVOLUTION METHOD INVOLVING FRACTIONAL DERIVATIVES

123

I. The Modified Operators of Fractional Calculus:

11. The Operators of the Modified Laplace Transformation and Their Inverse:

111. The Inverse Operators to the Generalized Stieltjes Transformation:

(~'{r(~)(l

+ x)-~)-'x-"

f)(x) = G:::

124

H.M. SRIVASTAVA, S.B. YAKUBOVICH, and Yu.F.LUCHKO

IV. T h e Operator with the Tricomi function 9 ( a 1b , x) in the Kernel and Its Inverse:


V. The Operator with Algebraic Function in the Kernel:

VI. T h e Operator with the Generalized Laguerre Polynomial L P ) ( z ) in the Kernel:

VII. The Inverse Operator to the Operator with the Confluent Hypergeometric Function l F l ( a ;6; -2) in the Kernel:

VIII. T h e Operator with the Gauss Hypergeometric Function F l ( a , b ; c; x) in the Kernel:

T H E CONVOLUTION M E T H O D INVOLVING FRACTIONAL DERIVATIVES

125

Definition 4 (Yakubovich [13]). Let f (x) E M,fYl(L) and g(x) E M,:,,(L). Then the G-convolution of the functions f ( x ) and g(x) is defined by

where the function Q k ( r ) (k = 1,2,3) has the form (5),


and the complex parameters aik)and pjk) satisfy the following conditions for the separation of poles for the contour from $ -ioo to $ +ico (see Definition 2 above):

Since the functions Q k ( r ) (k = 1 , 2 , 3 ) are the kernels of the corresponding G-transforms, we may set

and

,. = -/; + 1 73

$ ( ~ 3 -

83).

Remark 3. If we consider the following convolution:

then (by using the Eulerian integral representation for the Beta function and changing the order of integration) we readily obtain the usual Laplace convolution



126

for the functions f (z) and g(x). We will use the following result in our present investigation: Lemma 1. Let all A, B , C, a , b , c E W,i = f

l and

Then SUP

{~(U,V))

O

*I

+ + + + + +

+ + +

2sgn(A B) sgn(a b) 5 0, 2sgn(B C ) sgn(b c) 5 0, 2 sgn(C A) sgn(c a) 5 0, sgn(A B ) sgn(B C ) sgn(C A) 2sgn(a b c) 0. Lemma 1 can be proved by using elementary considerations, and we omit the details (see, for example, [2]). We state (without proof) our first result contained in

+ +

+ +

+ +

+ +

1

-1,

Re(&

+ /?I > - 11,

and 2sgn(cj)+sgn(yj-n)>0,

j=1,2;

n-k

k=O

provided t h a t n E No, a. # 0, Re(a) >

-4, a n d

provided t h a t Re(a - /?) > 0, Re(/?) < 0, and

(27)

130


provided t h a t


and

provided t h a t

and 2 s g n ( c j - i ) + s g n ( y j + R e { a - 6P) +- 7l ) 8.

20,

If H ( T ) = l/{r(rr+ ,f3 + r ) r ( l- a - ,l3 - T ) } , t h e n

j={i},

T H E CONVOLUTION M E T H O D INVOLVING FRACTIONAL DERIVATIVES

131

provided that 1 - - < Re(@+@) < 0 2 9. I f H(T) = l/{r(6 - cu

and

2sgn(cj - 1)

+ r)r(y -P -

T)},

+ sgn(yj) 2 0,

j = 1,2.

(38)

then


provided that

Remark 4. Each of the formulas (26), (31)) and (35) can easily be reduced to the known (classical) Leibniz rules for fractional integro-differentiatial operators. Remark 5. Many other Leibniz rules can similarly be obtained by applying other known summation theorems for generalized hypergeometric series (see, for example, Prudnikov et al. [7]). Finally, we have Theorem 4. Suppose that f E M&, (L), g E M;t,(L),

and @ y l ( r ) = H(T). Then each of the following integral analogues of Leibnzt rules holds true: 1. If H(T) = l/I'(a + /3 - 1 + T), then

provided that R e ( a + @ ) > 1 and

2sgn(cj-1)+sgn(yj)20,

j = 1,2.


132

2. If H ( T ) = F ( a

+ p + y + 6 + 1 - T ) / F ( P + 6 + 1 - T ) , then

( x - P - 6 ~ - a - 7 x a t P + r + 6 > ( f(.>s(x)) = ( f O t

* g)(x)

provided that

and

(

2sgn c j - - ~ ) + ~ ~ n ( 7 , - ~ e { ~ ~ : } ) > oj = . {i}.


3. If H ( T ) = r ( a

+ p + y + 6 - 1 + T ) , then

provided that

and

4.

2a+y+6 2 s g n ( c j ) + s g n ( y j - ~ e { ~ ~ + ~ + ~2}0) ,

If H ( r ) = l / { r ( a + 6 + 1 - r ) r ( P + y - 1 + T ) ) , then

provided that

i={t}

THE CONVOLUTION METHOD INVOLVING FRACTIONAL DERIVATIVES

and 2 sgn(cj - 1)

5. If H(T) = r(cr

+ 0-t r)/I'(cr

+ sgn(7j) 2 0, j = 1 , 2 . + @ +4 + T), t h e n

provided t h a t

and


2 ~ g n ( ~ j - l ) + ~ g n ( y j - l ) > O ,j = 1 , 2 .

Remark 6. The Leibniz rules and their integral analugues, considered in the present paper, have many important applications; in particular, in the evaluation of integrals and series with respect to the indices (parameters) of various classes of hypergeometric functions (see, for further details, Yakubovich and Luchko [15]).

ACKNOWLEDGEMENTS The present investigation was supported, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353. REFERENCES 1. Yu.A. Brychkov, H.4. Glaeske, and 0.1. Marichev, Factorization of interal transformations of convolution type, J. Soviet Math. 30 (1985), 2071-2094. 2. Nguen Thanh Hai and S.B. Yakubovich, The Double Mellin-Barnes Type Integrals and Their Applications to Convolution Theory, Series on Soviet and East European Mathematics, Vol. 6, World Scientific Publishing Company, Singapore, New Jersey, London, and Hong Kong, 1992. 3. Yu.F. Luchko and S.B.Yakubovich, Generating operators and convolutions for some integral transforms, Dokl. Akad. Nauk BSSR 35 (1991), 773-776. 4. 0.1. Marichev, Handbook of Integral Transforms of Higher Transcendental Functions: Theory and A Igorithmic Tables, translated from the Russian by L.W. Longdon, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Brisbane, Chichester, and Toronto, 1983. 5. T.J. Osler, A n integral analogue of Taylor'a series and its use in computing Fourier transforma, Math. Comput. 26 (1972), 449-460. 6. T.J. Osler, The integral analog of the Leibniz rule, Math. Comput. 26 (1972), 903-915. 7. A.P. Prudnikov, Yu.A. Brychkov, and 0.1. Marichev, Integrals and Series, Vols. 1 , 2 and 3, Gordon and Breach Science Publishers, New York, 1986, 1986, and 1989.

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H.M. SRIVASTAVA, S.B. YAKUBOVICH, and Y u F . LUCHKO

8 . M. Saigo and S.B. Yakubovich, O n the theory of convohtion integrals for G - t r a n s f o r m s , Fukuoka Univ. Sci. Rep. 2 1 (1991), 181-193. 9 . S.G. Samko, A.A. Kilbas, and 0.1. Marichev, Integrals and Derivatives of Fractional Order and S o m e of Their Applications, Nauka i Tekhnika, Minsk, 1987. 10. H.M. Srivastava, K.C. Gupta, and S.P. Goyal, T h e H-Functions of One and T w o Variables with Applications, South Asian Publishers, New Delhi and Madras, 1982. 11. Vu Kim Tuan, 0.1. Marichev, and S.B. Yakubovich, Composition structures of integral transformations, Soviet Math. Dokl. 33 (1986), 166-170. 12. Y. Watanabe, Notes on the generalized derivative of Riemann-Liouville and its application to Leibniz's formula. I end 11, TBhoku Math. J . 34 (1931), 8-27 and 28-41. 13. S.B. Yakubovich, O n the constructive method of the building of the integral convolutions, Dokl. Akad. Nauk BSSR 34 (1990), 588-591. 14. S.B. Yakubovich and Yu.F. Luchko, T h e generalization of Leibniz rule on the integral convolutions, Dokl. Akad. Nauk BSSR 35 (1991), 111-115. 15. S.B. Yakubovich andYu.F. Luchko, T h e evaluation of integrals and series with respect to in-


dices (parameters) of hypergeometric functions, Proceeding of the International Symposium on Symbolic and Algebraic Computation, Bonn, July 1991, pp. 271-280.

Integral Transforms and Special Functions The ...

Integral Transforms and Special Functions The ...

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