The Mittag-Leffler (M-L) function [13] is defined as. z k Γ(kα + 1) (α > 0). A further, two-index generalization of this function is given as

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1 M a t h e m a t i c a B a l k a n i c a New Series Vol. 21, 2007, Fasc. 3-4 On Mittag-Leffler Type Function and Fractional Calculus Operators 1 Mridula Garg a, Alka Rao a and S.L. Kalla b Presented by V. Kiryakova The aim of this paper is to study some properties of Mittag-Leffler type function E σ,δ,ρ (z introduced by Kilbas and Saigo. It is an entire function and arises in the solution of some linear Abel-Volterra integral equations. Here we establish two theorems which provide the image of this function under the fractional integral operators involving Fox H-function. The other results obtained are the images under the fractional integral and differential operators defined by Saigo and Erdélyi-Kober respectively. Some known special cases have also been mentioned. AMS Subj.Classification: 26A33, 33C60, 33E12 Key Words: Fractional calculus operators, H-function, Mittag-Leffler function 1. Introduction The Mittag-Leffler (M-L function 13] is defined as (1.1 E α (z = k=0 z k Γ(kα + 1 (α > 0. A further, two-index generalization of this function is given as (1.2 E α,β (z = k=0 z k Γ(kα + β (α > 0, β > 0, Mittag-Leffler 14] and Wiman 21] have investigated properties of M-L function (1.1 while the function (1.2 was introduced and studied by Wiman 21] and Dedicated to 70-th anniversary of Prof. Megumi Saigo (Fukuoka University, Japan, May

2 350 M. Garg, A. Rao and S.L. Kalla later by Agarwal 1], Humbert and Agarwal 5] and by Dzrbashjan in his book 2]. Kiryakova 10,11] has introduced and studied a multiindex Mittag- Leffler function as an extension of the generalized Mittag-Leffler function considered by Dzrbashjan. The functions (1.1 and (1.2 were also studied by many other authors. A detailed account of these two functions is given in 3]. Since the M-L function provides solutions to certain problems formulated in terms of fractional order differential, integral and difference equations, it has recently become a subject of interest for many authors in the field of fractional calculus and its applications. A further extension of this function has been introduced by Kilbas and Saigo 8] as follows (1.3 E σ,δ,ρ (z = c p z p, with c p = p 1 i=0 Γσ(iδ + ρ + 1] (n = 0, 1, 2,, where σ > 0, δ > 0, σ(iδ + ρ 1, 2, 3 (i = 0, 1, 2, and an empty product is interpreted as unity. In another paper Kilbas and Saigo 9] have established some properties and the explicit formulas for Riemann-Liouville fractional integral and derivative for this function. Gorenflo, Kilbas and Rosogin 4] have also studied some other properties of such Mittag- Leffler type function. Throughout the present work the following conditions will be considered as existence conditions for the above function: (1.4 σ > 0, δ > 0 and ρ > 1 σ. Obviously, the function (1.3 is a generalization of the function (1.2, obtained by replacing δ = 1 in (1.3: (1.5 E σ,1,ρ (z = Γ(σρ + 1E σ,σρ+1 (z and for δ = 1 and ρ = 0, (1.3 reduces to (1.1. In the present paper our aim is to study the generalized Mittag-Leffler type function given by (1.3. We find the image of this function under the fractional integral operators involving Fox H-function defined by Kalla 6,7] and

3 On Mittag-Leffler Type Function and further studied by Srivastava and Buschman 19]. We use the following notations for the left-sided and right-sided generalized fractional integral operators (1.6 R ζ 1,ζ 0+ f(t](x = x ζ 1 ζ 1 x 0 t ζ 1 (x t ζ f(th M,N P,Q λu (a j, α j 1,P (b k, β k 1,Q dt, (1.7 (a j, α j 1,P R ζ 2,ζ f(t](x = xζ 2 t ζ2 ζ 1 (t x ζ f(th M,N P,Q λv dt, x (b k, β k 1,Q ( t m ( where U and V represent the expressions 1 t n ( x m ( and 1 x x x n t t respectively, with m, n > 0. Here H M,N P,Q stands for well known Fox H-function, defined by means of the following Mellin-Barnes type integral 20] (a j, α j 1,P (1.8 H M,N P,Q z = 1 θ(sz s ds 2πi (b k, β k L 1,Q where (1.9 θ(s = M Γ(b k β k s N Γ(1 a j + α j s k=1 Q k=m+1 j=1 Γ(1 b k + β k s P j=n+1 Γ(a j α j s and L is a suitable contour in C, the orders (M, N, P, Q are integers, 1 M Q 0 N P and the parameters a j, b k R; α j > 0, j = 1,,P, β k > 0, k = 1,,Q are such that α j (b k + l β k (a j l 1, l, l = 0, 1, 2,. For various type of contours and the conditions for existence, analyticity of the H-function and other details one can see 12,15,20]. We assume that these conditions are satisfied by H-function throughout the present work. 2. The image of the Mittag-Leffler type function under the left-sided fractional integral operator Theorem 1. Let σ > 0, δ > 0, ρ > 1 σ and Rζ 1,ζ 0+ be the generalized left-sided fractional integral operator (1.6, then the following result holds:

4 352 M. Garg, A. Rao and S.L. Kalla R ζ 1,ζ 0+ tζ E σ,δ,ρ (at ν ](x = x ξ ( p 1 Γσ(iδ + ρ + 1] (ax ν p i=0 (2.1 H M,N+2 P+2,Q+1 λ ( ζ 1 ξ νp, m, ( ζ, n, (a j, α j 1,P (b k, β k 1,Q ( 1 ζ 1 ζ ξ νp, m + n, The conditions for validity of (2.1 are (i min(ζ, ζ 1, ξ, ν, m, n > 0 (ii ζ 1 + ξ + m min 1 k M bk β k ] + 1 > 0 ] bk (iii ζ + n min + 1 > 0. 1 k M β k (iv A > 0, arg(λ < Aπ 2, where (2.3 A = N P M Q α j α j + β k j=1 j=n+1 k=1 k=m+1 β k P roof. Using the definition (1.6 in the left hand side of (2.1, writing the functions in the forms given by (1.3 and (1.8, interchanging the order of integrations and summation and evaluating the t-integral as beta integral, we easily arrive at the result (2.1 under the conditions ( Images under the left-sided fractional calculus operators defined by Saigo If in Theorem 1 we choose the parameters as follows: M = 1, N = P = Q = 2, a 1 = 1 α β, a 2 = 1 + η, b 1 = 0, (3.1 b 2 = 1 α, α 1 = α 2 = β 1 = β 2 = 1, m = 0, n = 1, λ = 1, then the Fox H-function reduces to the Gauss hypergeometric function 20, p.40] and the fractional integral operator R ζ,ζ 0+ reduces to the left-sided fractional

5 On Mittag-Leffler Type Function and integral operator defined by Saigo 16, eqn. (1.1]. The two operators are connected by the formula (3.2 R 0,α 1 0+ f Γ(α + β Γ( η x β I α,β,η 0+ f where (3.3 I α,β,η 0+ f = x α β Γ(α x 0 (x t α 1 F ( α + β, η; α; 1 t f(tdt α > 0, β, η R x Saigo 17, p.117] has given the following definition for fractional differential operator, corresponding to fractional integration (3.3: (3.4 I α,β,η 0+ f Dn xi α+n,β n,η n 0+ f for α 0, 0 < α + n 1 (n is a positive integer however, we prefer to use the following notation for the fractional differential operator (of non negative order: (3.5 D α,β,η 0+ f Dn xi n α,β n,η n 0+ f for α 0, 0 < n α 1 (n is a positive integer Now, we find the image of the Mittag-Leffler type function under the left-sided fractional integral and differential operators defined by Saigo. Corollary 1. Let the conditions (1.4 for existence of the function E σ,δ,ρ (z be satisfied, then for α > 0, η > β, ν > 0, ξ > 1 we get I α,β,η 0+ tξ E σ,δ,ρ (at ν ](x = x ξ β ( p 1 Γσ(iδ + ρ + 1] i=1 (3.6 (3.7 Γ(1 + ξ + νp Γ(1 + ξ + η β + νp Γ(1 + ξ β + νp Γ(1 + α + ξ + η + νp (axν p D α,β,η 0+ tξ E σ,δ,ρ (at ν ](x = x ξ β Γ(1 + ξ + νp Γ(1 + ξ β + νp ( p 1 Γσ(iδ + ρ + 1] i=1 Γ(1 + ξ + η β + νp Γ(1 + ξ α + η + νp (axν p.

6 354 M. Garg, A. Rao and S.L. Kalla P roof. The result (3.6 follows from Theorem 1, if we make substitutions as given in (3.1 and use the relation (3.2. We first express the fractional derivative of the function in the following form, using (3.5 (3.8 D α,β,η 0+ tξ E σ,δ,ρ (at ν ](x = D n x I n α,β n,η n 0+ t ξ E σ,δ,ρ (at ν ]. Applying the result (3.6 and the well-known formula for the x-th derivative of power function (3.9 D µ (x λ = Γ(λ + 1 Γ(λ µ + 1 xλ µ, λ > 1 we get the required result (3.7 after a little simplification. If in the above result we take α+β = 0, the fractional calculus operators defined by Saigo reduce to the left-sided Riemann-Liouville fractional calculus operators. Further on giving particular values to the parameters, we get the results obtained recently by Kilbas and Saigo 9, Th.2, Th.3]. 4. Images under the left-sided fractional calculus operators defined by Erdélyi-Kober If we take β = 0 in the Saigo integral operator (3.3, we get the left-sided fractional integral operator of Erdélyi-Kober (E-K, see16], (4.1 I α,0,η 0+ f = Eα,η 0+ f, defined as follows: (4.2 E α,η 0+ f = x α η Γ(α x 0 (x t α 1 t η f(tdt, α > 0. The definition of the corresponding E-K fractional differential operator is given as 18, p.322] (4.3 E α,η 0+ f = x α η D n xx n+α+η E n+α,η f] for α 0, 0 < n + α 1; n is a positive integer.

7 On Mittag-Leffler Type Function and however we use following notation for the left-sided fractional differential operator: (4.4 D α,η 0+ f = xα η D n xx n α+η E n α,η f] for α 0, 0 < n α 1; n is a positive integer. We now give images of the Mittag-Leffler type function under the leftsided fractional integral and differential operators of Erdélyi-Kober. If we make suitable choice of the parameters, we get the following interesting results giving images of the Mittag-Leffler function in terms of the same function. Corollary 2. Let the conditions (1.4 for existence of the function Eα, δ, ρ(z be satisfied, then ] (4.5 E σ,σρ 0+ E σ,δ,ρ (at σδ (x = 1 ]] x σδ E σ,δ,ρ (ax σδ 1 a ] t σρ E σ,δ,ρ (at σδ (x D σ,σ σδ 0+ (4.6 = Γσ(ρ δ Γσ(ρ δ + 1] x σρ + ax σ(ρ+δ E σ,δ,ρ (ax σδ P roof. On taking in (3.6 and using the relation (4.1, we easily arrive at the following result ( p 1 E α,η 0+ tξ E σ,δ,ρ (at ν ](x = x ξ Γσ(iδ + ρ + 1] (4.7 i=1 Γ(1 + ξ + η + νp Γ(1 + α + ξ + η + νp (axν p. Further, taking α = σ, η = σ, ξ = 0, ν = σδ and doing some manipulations we arrive at the first assertion (4.5. Now applying the fractional differential operator (4.4 to the Mittag- Leffler type function and proceeding on the lines of the proof as given in Corollary 1 we easily arrive at the following result D α,η 0+ tξ E σ,δ,ρ (at ν ](x = x ξ ( p 1 i=1 Γσ(iδ + ρ + 1] Γ(1 + ξ + η + νp (4.8 Γ(1 + ξ α + η + νp (axν p. On specializing the parameters, we obtain the second assertion (4.6.

8 356 M. Garg, A. Rao and S.L. Kalla 5. Image of the Mittag-Leffler type function under the rightsided fractional integral operator Theorem 2. Let σ > 0, δ > 0, ρ > 1 σ and Rζ 2,ζ be the right-sided fractional integral operator (1.7 then the following result holds: R ζ 2,ζ t ξ E σ,δ,ρ (at ν ](x = x ξ q=0 ( q 1 Γσ(iδ + ρ + 1] (ax ν q i=0 (5.1 H M,N+2 P+2,Q+1 λ ( ζ 2 ξ νq, m, ( ζ, n, (a j, α j 1,P (b k, β k 1,Q ( ζ 2 ζ ξ νp, m + n, The conditions for validity of (5.1 are (i min(ζ, ζ 2, ξ, ν, m, n > 0 (ii ζ + n min 1 k M bk β k ] + 1 > 0 bk (iii ζ 2 + ξ + m min 1 k M β k ] > 0. (iv A > 0, argλ < Aπ 2, where (5.2 A = N P M Q α j α j + β k j=1 j=n+1 k=1 k=m+1 β k P roof. The result can easily be established on the lines similar to that of Theorem 1. Saigo 6. Images under the right-sided fractional calculus operators of In Theorem 2, if we choose the parameters as given in (3.1, the fractional integral operatorr ζ 2,ζ reduces to the other fractional integral operator Jx α,β,η defined by Saigo 16, eq.(1.3]. For uniformity, we shall term it as right-sided fractional integral operator of Saigo, and represent it as follows:

9 On Mittag-Leffler Type Function and For α > 0, and real numbers β, η (6.1 I α,β,η f = 1 Γ(α x 0 (t x α 1 t α β F The two operators are connected by the formula (6.2 R β,α 1 Γ(α + βγ( ηi α,β,η ( α + β, η; α; 1 x f(tdt t Further, Saigo has represented the corresponding fractional differential operator as 17, p.117] (6.3 I α,β,η f ( 1 n DxI n α+n,β n,η f α 0, 0 < α + n 1, (n is a positive integer, however, we prefer to use the following notation for the Saigo fractional differential operator: (6.4 D α,β,η f ( 1 n DxI n n α,β n,η f; α 0, 0 < n α 1, (n is a positive integer. Corollary 3. Let the conditions (1.4 for existence of the function E σ,δ,ρ (z be satisfied, then for α > 0, η > ξ, ν > 0 the following results hold: I α,β,η t ξ E σ,δ,ρ (at ν ](x ( p 1 = x ξ β Γσ(iδ + ρ + 1] i=0 Γ(β + ξ + νp (6.5 Γ(ξ + νp D α,β,η t ξ E σ,δ,ρ (at ν Γ(β + ξ + νp (6.6 Γ(ξ + νp Γ(η + ξ + νp Γ(α + β + η + ξ + νp (ax ν p ](x = x ξ β ( p 1 Γσ(iδ + ρ + 1] i=0 Γ(η + ξ + νp Γ(β α + η + ξ + νp (ax ν p where n is a positive integer such that 0 < n α 1. P roof. The proof goes in lines similar to this in the case of left-sided operators.

10 358 M. Garg, A. Rao and S.L. Kalla If in the above results we take α + β = 0, the fractional calculus operators defined by Saigo reduce to the right-sided fractional calculus operators of Riemann-Liouville. Further on, giving particular values to the parameters, we get the results obtained recently by Kilbas and Saigo 9, Th. 3 and Th. 5]. 7. Images under the right-sided Erdélyi-Kober fractional calculus operators Taking β = 0 in the operator (6.1, we get the right-sided Erdélyi-Kober fractional integral operator 16] (7.1 I α,0,η 0+ f = Eα,η f, α > 0, where (7.2 E α,η f = xη Γ(α x 0 (t x α 1 t α n f(tdt; α > 0. The definition of the right-sided Erdélyi-Kober fractional differential operator is given as 18] (n is a positive integer ] (7.3 E α,η f = xη ( D n x x n η E n+α,η n f α 0, 0 < n + α 1 but we use the following notation for the Erdélyi-Kober fractional differential operator (n is a positive integer ] (7.4 D α,η f = xη ( D n x x n η E n α,η n f α 0, 0 < n α 1 Corollary 4. following results hold: For α > 0, η > ξ, ν > 0 and σ, δ > 0, ρ > 1 σ the ] (7.5 E σ,σρ+1 E σ,δ,ρ (at σδ (x = 1 ] a xσδ E σ,δ,ρ (ax σδ 1, a 0 ] D σ, σδ t σ(ρ+1 1 E σ,δ,ρ (at σδ (x (7.6 = Γσ(ρ δ ] Γσ(ρ δ + 1] x σ(ρ ax σ(ρ+δ+1 1 E σ,δ,ρ (ax σδ

11 On Mittag-Leffler Type Function and P roof. The proof follows on the lines similar to this in the case of left-sided operators. Acknowledgement: The authors are thankful to the worthy referees for their valuable suggestions which led to the present form of the paper. The second author is thankful to UGC, New Delhi for awarding Teacher Research Fellowship. References 1] R.P. A g a r wal, A propos d une note M. Pierre Humbert, Acad. Sc. Paris, 236(1953, ] M. M. D zrbashajan, Integral Transforms and Representations of Functions in the Complex Domain (in Russian, Nauka, Moscow, ] A. E r délyi, W. M a g nus, F. Oberhettinger a n d F. G. T r i c o m i, Higher Transcendental Functions, Vol. 3, McGraw-Hill, New York- Toronto-London, ] R. G o r enflo, A. A. Kilbas a n d Rosogin, Mittag-Leffler type functions, Integral Transforms and Special Functions, 7 (1998, No.3-4, ] A. H u m bert a nd R.P. A g a r wal, Sur la fonction de Mittag-Leffler et quelques unes de ses generalizations, Bulle. Sci. Math., 77 (1953,(2, ] S.L. K a l la, Integral operators invoving Fox s H-function, Acta Mexicana Cien. Tec., 3,(1969, ] S.L. K a lla, Integral operators invoving Fox s H-function II, Nota Cien. Tec., 3(1969, ] A.A. Kilbas, M. Saigo, On solution of integral equation of Abel- Volterra type, Differential and Integral Equations, 5 (1995, ] A. A. K i l bas, M. S a i g o, On Mittag-Leffler type function, fractional calculus operators and solutions of integral equations, Integral Transforms and Special Functions, 4 (1996,( ] V. K iryakova, Multiindex Mittag-Leffler functions, related Gelfond- Leontiev operators and Laplace type integral transforms, Fractional Calculus and Appl. Analysis, 2 (1999, No 4, ] V. K i ryakova, Multiple (multiindex Mittag-Leffler functions and relations to generalized fractional calculus, J. Comput. Appl. Math., 118(2000, ] A. M. M a thai a nd R.K. Saxena, The H-Function with Applications in Statistics and Other Disciplines, Wiley Eastern Ltd., New Delhi and John Wiley and Sons Inc, New-York, 1978.

12 360 M. Garg, A. Rao and S.L. Kalla 13] G. M. M i ttag-leffler, Sur la nouvelle function E α (x, C.R. Acad. Sci. Paris, 137 (1903, ] G. M. M ittag-leffler, Sur la representation analytique d une branche uniforme une function monogene, Acta. Math., 29(1905, ] A. P. P r udnikov, Y u. A. Brychkov, O.I. M a r i c hev, Integrals and Series. More Special Functions, Gordon and Breach Sc. Publ., New York, ] M. Saigo, A remark on integral operators involving the Gauss hypergeometric Functions, Math. Rep. Kyushu Univ., 11(1978, ] M.Saigo, A certain boundary value problem for the Euler-Darboux Equations III, Mathematica Japonica, 26 (1981, (1, ] S.G. Samko, A. A. K i lbas a n d O.I. M a richev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach Sc. Publ., New York, ] H. M. S r ivastava a n d R.G. B uschman, Integral Operators invoving Fox s H-function Acta Mexicana Cien. Tec., 7(1973, ] H.M. S r ivastava, K. C. G u pta a nd S.P. G o y a l, The H- Functions on One and Two Variables, South Asian Publishers, New-Delhi, ] A. W i m a n, Uber den Fundamental satz in der Theorie de Functionen, Acta Math, 29 (1905, a Department of Mathematics, Received University of Rajasthan, Jaipur , INDIA s: gargmridula@gmail.com alkarao.ar@rediffmail.com b Department of Mathematics and Computer Science, Kuwait University, P.O.Box-5969, Safat 13060, KUWAIT shyamkalla@gmail.com