FRACTIONAL CALCULUS FORMULAS INVOLVING H-FUNCTION AND SRIVASTAVA POLYNOMIALS

Commun. Korean Math. Soc. 31 2016 No. 4 pp. 827 844 http://dx.doi.org/10.4134/ckms.c150251 pissn: 1225-1763 / eissn: 2234-3024 FRACTIONAL CALCULUS FORMULAS INVOLVING H-FUNCTION AND SRIVASTAVA POLYNOMIALS Dinesh Kumar Abstract. Here in this paper we aim at establishing some new unified integral and differential formulas associated with the H-function. Each of these formula involves a product of the H-function and Srivastava polynomials with essentially arbitrary coefficients and the results are obtained in terms of two variables H-function. By assigning suitably special values to these coefficients the main results can be reduced to the corresponding integral formulas involving the classical orthogonal polynomials including for example Hermite Jacobi Legendre and Laguerre polynomials. Furthermore the H-function occurring in each of main results can be reduced under various special cases. 1. Introduction and preliminaries The Fox s H-function is defined and represented in the following manner [13: [ 1.1 Hpq mn z ajαj 1p b j 1 θs z s ds 1q 2πi L where i 1 and 1.2 θs m j1 Γb j s n j1 Γ1a j +α j s q jm+1 Γ1b j + s p jn+1 Γa j α j s. The nature of the contour L of the integral 1.1 the conditions of existence of the H-function defined by 1.1 and other details can be found in the book Kilbas and Saigo [8. A lot of research work has been recently come up on the study and development of a function that is more general than Fox s H-function known as H-function. The H-function was introduced by Inayat-Hussain [6 7 which is Received December 31 2015; Revised March 24 2016. 2010 Mathematics Suect Classification. 26A33 33C45 33C60 33C70. Key words and phrases. generalized fractional calculus operators fractional calculus H- function of two variables Srivastava polynomials special function. 827 c 2016 Korean Mathematical Society

828 DINESH KUMAR a generlaization of familiar Fox H-function it is defined and represented in the following manner [2: 1.3 H MN PQ [z HMN PQ [ z ajαj;aj 1N ajαj N+1P b j 1M b j;b j M+1Q 1 2πi L θξ z ξ dξ z 0 where i 1 and M j1 1.4 θξ Γb j ξ N j1 {Γ1a j +α j ξ} Aj Q jm+1 {Γ1b j + ξ} Bj P jn+1 Γa j α j ξ. It may be noted that the θξ contains fractional powers of some of the gamma function and MNPQ are integers such that 1 M Q 1 N P. Also a j 1P and b j 1Q are complex parameters; α j 1P 1Q are positive real numbers not all zero simultaneously and the exponents A j 1N and B j M+1Q may take non-integer values which we assume to be positive for standardization purpose. The nature of contour L sufficient conditions of convergence of defining integral 1.4 and other details about the H-function reader can also refer the papers [6 7 10 for more detail. The behavior of the H-function for small values of z and sufficient conditions for the absolute convergence of the defining integral follows easily [2 17 1.5 H MN PQ [z o z α where 1.6 α min 1 j M Re M N 1.7 Ω + A j α j j1 j1 Q jm+1 z 0 B j P jn+1 α j > 0 and arg z < 1 2 Ωπ. Recently the H-function of two variable is defined and represented by Singh and Mandia [27 in the following manner: [ x H[xy H y 1.8 [ H 0n1:m2n2:m3n3 x p 1q 1:p 2q 2:p 3q 3 y 1 4π 2 θ 1 ξµθ 2 ξθ 3 µx ξ y µ dξdµ xy 0 L 2 L 1 ajαj;aj 1p1 cjγj;kj 1n2 cjγj n2 +1p ejηj;rj 2 1n3 ejηj n3 +1p 3 βj;bj 1q1 djδj 1m2 djδj;lj m2 +1q fjλj 2 1m3 fjλj;sj m3 +1q 3 1.9 θ 1 ξµ n1 j1 Γ1a j +α j ξ +A j µ p1 jn Γa 1+1 j α j ξ A j µ q 1 j1 Γ1b j + ξ +B j µ

FRACTIONAL CALCULUS FORMULAS 829 1.10 θ 2 ξ n2 j1 {Γ1c j +γ j ξ} Kj m 2 j1 Γd j δ j ξ p2 jn Γc 2+1 j γ j ξ q 2 jm {Γ1d 2+1 j +δ j ξ} Lj 1.11 θ 3 µ n3 j1 {Γ1e j +η j µ} Rj m 3 j1 Γf j λ j µ p3 jn Γe 3+1 j η j µ q 3 jm {Γ1f 3+1 j +λ j µ} Sj ThegeneralclassofpolynomialsSrivastavapolynomialsS m n [x willbedefined and represented as follows [28 p. 1 eq. 1: 1.12 S m n [x n mk A nk x k n 012... where m is an arbitrary positive integer and the coefficient A nk nk 0 are arbitrary constants real or complex. Where λ n denotes the Pochhammer symbol defined by 1.13 γ n Γγ +n Γγ { 1 n 0γ 0 γγ +1 γ +n1 n Nγ C By suitable specializing the coefficient A nk the polynomials Sn m x can easily be reduced to the classical orthogonal polynomials including for example the Hermite polynomials H n x the Jacobi polynomials P n αβ x and the Laguerre polynomials L α n x and also to several familiar particular cases of the Jacobi polynomials such as the Gegenbauer polynomials Cn ν x the Legendre polynomials P n x and the Tchebycheff polynomials T n x and U n x see for detail [30. Other interesting special cases of the Srivastava polynomials 1.12 include such generalized hypergeometric polynomials as the Bessel polynomials the generalized Hermite polynomials and the Brafman polynomials as a particular case. 2. Generalized fractional differintegral operators If αα ββ γ C and [Reγ > 0 x > 0 then the generalized fractional calculus operators involving Appell function F 3 given by Saigo and Maeda [20 are defined by I αα ββ γ 2.1 0+ f x xα Γγ d dx 0 x t α xt γ1 F 3 αα ββ ;γ; 1 t x 1 x t k I αα β+kβ γ+k 0+ f I αα ββ γ f x ftdt x Reγ 0; k [Reγ+1

830 DINESH KUMAR 2.2 2.3 2.4 xα t α tx γ1 F 3 αα ββ ;γ; 1 x Γγ x t 1 t ftdt x d k I αα ββ +kγ+k f x Reγ 0; k [Reγ+1 dx D αα ββ γ 0+ f x I α αβ βγ 0+ f d dx 1 Γnγ x Reγ > 0 k I α αβ +kβγ+k 0+ f d dx n x x α xt nγ1 t α F 3 α αnβ βnγ;1 t x 1 x t D αα ββ γ f x I α αβ βγ d dx 1 Γnγ 0 f x Reγ > 0 k I α αβ β+kγ+k d dx n x α tx nγ1 t α F 3 α αβ nβnγ; 1 x t 1 t x x x Reγ > 0; k [Reγ+1 ftdt f x Reγ > 0; k [Reγ+1 ftdt. Here F 3 αα ββ ;γ;zξ is the familiar Appell hypergeometric function of two variables defined by 2.5 F 3 αα ββ ;γ;zξ m0n0 α m α n β m β n γ m+n z m ξ n m! n! where λ n denotes the Pochhammer symbol defined by { λλ+1 λ+n1 n N λ n 1 n 0. z < 1 and ξ < 1 It is noted that the series in 2.5 is absolutely convergent for all z ξ C with z < 1 and ξ < 1 and for all z ξ C\{1} with z 1 and ξ 1. The generalized fractional calculus operators due to Saigo-Maeda defined in [3 12 16 21 25 reduce to the following generalized fractional calculus

FRACTIONAL CALCULUS FORMULAS 831 operators due to Saigo [18 24 26: 2.6 I α0ββ γ 0+ f x 2.7 2.8 2.9 I α0ββ γ f x I γαγβ 0+ f I γαγβ x γ C x > 0 f x γ C x > 0 D 0α ββ γ 0+ f x D γα γβ γ 0+ f x Reγ > 0 x > 0 D 0α ββ γ f x D γα γβ γ Further from [20 p. 394 eq. 4.18 and 4.19 we have: f x Reγ > 0 x > 0. Lemma 1. The next formula holds for αα ββ γ C Reα > 0: I αα ββ γ 0+ t ρ1 x 2.10 ΓρΓρ+γ αα βγρ+β α Γρ+γ αα Γρ+γ α βγρ+β xραα +γ1 where Reγ > 0 Reρ > max[0reα+α +β γreα β x > 0. In particular if α 0 β η γ α and α is replaced by α+β in 2.10 then Saigo-Maeda operator becomes Saigo operator and we obtain [20 2.11 I αβη 0+ tρ1 x ΓρΓρ+η β Γρβ Γρ+α+η xρβ1 x > 0. Lemma 2. The next formula holds for αα ββ γ C Reα > 0: 2.12 I αα ββ γ t ρ1 x where Γ1+α+α γ ρ Γ1+α+β γργ1βρ Γ1ρΓ1+α+α +β γργ1+αβρ x ραα +γ1 Reγ > 0x > 0Reρ < 1+min[ReβReα+α γreα+β γ. In particular if α 0 β η γ α α is replaced by α + β in 2.12 then 2.13 I αβη t ρ1 Γ1+β ργ1+η ρ x Γ1ρΓ1+α+β +η ρ xρβ1 x > 0 Lemma 3. The next formula holds for αα ββ γ C Reα > 0: D αα ββ γ 0+ t ρ1 x 2.14 ΓρΓρ+αβΓρ+α+α +β γ γ1 ΓρβΓρ+α+α γγρ+α+β γ xρ+α+α where Reρ > 0 Reρ+αβ > 0 Reρ+α+α +β γ > 0 x > 0.

832 DINESH KUMAR Lemma 4. The next formula holds for αα ββ γ C Reα > 0: 2.15 D αα ββ γ t ρ1 x 1 n Γ1αα +γργ1α β+γργ1+β ρ Γ1ρΓ1αα β+γργ1α +β ρ xρ+α+α γ1 where Re1+β ρ > 0 Re1ρα β +γ > 0 Re1ραα +γ > 0 n [Reγ+1 x > 0. 3. Fractional differintegral formulas In this section we will establish four fractional differintegral formulas for the product of H-function and Srivastava polynomials. Theorem 1. Let αα ββ γµηδυzab C λσ > 0 and Reγ > 0. Then we have { I αα ββ γ 0+ t µ1 bat η Sn [t m λ bat δ H MN PQ [ } zt σ bat υ a j α j ;A j 1N a j α j N+1P x b j 1M b j ;B j M+1Q b η x µαα +γ1 n mk A nk b δk x λk [ H 04:MN:10 zx σ b υ 3.1 44:PQ:01 where E 1 1η δkυ;11µλkσ;1 E 1 a j α j ;A j 1N a j α j N+1P E 2 b j 1M b j ;B j M+1Q 01;1 1µγ +α+α +β λkσ;11µ+α β λkσ;1;and E 2 1η δkυ;01µγ +α+α λkσ;1 1µγ +α +β λkσ;11µβ λkσ;1. Also satisfy the following conditions: 3.2 i arg z < 1 M 2 Ωπ Ω > 0 Ω + ii a b x < 1 also we have j1 N A j α j j1 Q jm+1 B j P jn+1 Reµ+σ min 1 j M Re > max[0reα β Reα+α +β γ Reη+υ min 1 j M Re > max[0reα β Reα+α +β γ. α j.

FRACTIONAL CALCULUS FORMULAS 833 Proof. To prove the fractional integral formula3.1 we first express the general class of polynomials occurring on its left-hand side in the series from given by 1.12 replace the H-function occurring therein by its well-known Mellin- Barnes contour integral given by 1.3 interchange the order of summations and make a little simplification. 3.3 I 1 Iαα ββ γ 0+ n mk 1 A nk θsz s ds 2πi L 1 } x. b ηδkυs 1 a b t η+δk+υs t µ+λk+σs1 Now interchanging the order of integrals and summations we obtain the following form after a little simplification: 2 I 1 b η n mk 1 A nk b δk θsz s b υs a ξ dsdξ 2πi L 1 L 2 b { I αα ββ γ 0+ t µ+λk+σs+ξ1 3.4 } x 3.5 Γη+δk+υs+ξ Γη+δk+υsΓ1+ξ b η x µαα +γ1 1 1 2πi L 2 Γ1+ξ n mk ξ dξ Γµ+λk+σs+γαα β+ξγµ+λk+σsα +β +ξ Γµ+λk+σs+γα β+ξγµ+λk+σs+β +ξ A nk b δk x λk 1 θs zb υ x σ s ds 2πi L 1 Γη+δk+υs+ξΓµ+λk+σs+ξ Γη+δk+υsΓµ+λk+σs+γαα +ξ by re-interpreting the Mellin-Barnes counter integral in terms of the H-function of two variables defined by 1.8 we obtain the right-hand side of 3.1 after little simplifications. This completes proof of Theorem 1. In view of the relation 2.6 then we get the following corollary concerning left-sided Saigo fractional integral operator [18 19 20. Corollary 1.1. Let αβγµηδυzab C; λσ > 0 and Reα > 0. Then the following result holds true: { I αβγ 0+ t µ1 bat η Sn [t m λ bat δ [ } H MN PQ zt σ bat υ a j α j ;A j 1N a j α j N+1P x b j 1M b j ;B j M+1Q 3.6 b η x µβ1 H 03:MN:10 33:PQ:01 n mk A nk b δk x λk [ zx σ b υ E 1 a jα j ;A j 1N a j α j N+1P E 2 b j 1M b j ;B j M+1Q 01;1

834 DINESH KUMAR where E 1 1η δkυ;11µλkσ;1 1µλk +β γσ;1; and E 2 1η δkυ;01µλk +βσ;1 1µλk αγσ;1. Also satisfy the following conditions: Reµ+σ min 1 j M Re > max[0reβ γ Reη+υ min 1 j M Re > max[0reβ γ the conditions i and ii given in Theorem 1 are also satisfied. Next if we set β α in 3.6 we obtain the following result concerning left-sided Riemann-Liouville fractional integral operator [14 18: Corollary 1.2. Let αµηδυzab C; λσ > 0 and Reα > 0. Then we have { I0+ α t µ1 bat η Sn [t m λ bat δ 3.7 H MN PQ b η x µ+α1 [ } zt σ bat υ a j α j ;A j 1N a j α j N+1P x b j 1M b j ;B j M+1Q H 02:MN:10 22:PQ:01 n mk A nk b δk x λk [ zx σ b υ E 1a j α j ;A j 1N a j α j N+1P E 2b j 1M b j ;B j M+1Q 01;1 where E 1 1η δkυ;11µλkσ;1; and E 2 1η δkυ;0 1µαλkσ;1 and the existence conditions of the above corollary easily follows with the help of 3.6. Theorem 2. Let αα ββ γµηδυzab C λσ > 0 and Reγ > 0. Then we have { I αα ββ γ t µ1 bat η Sn [t m λ bat δ H MN PQ [ } zt σ bat υ a j α j ;A j 1N a j α j N+1P x b j 1M b j ;B j M+1Q b η x µαα +γ1 n mk A nk b δk x λk [ H 04:MN:10 zx σ b υ 3.8 44:PQ:01 where F 1 a j α j ;A j 1N a j α j N+1P F 2 b j 1M b j ;B j M+1Q 01;1 F 1 1η δkυ;1µ+λk +γ αα σ;1 µ+γ αβ +λkσ;1µ+β +λkσ;1;and F 2 1η δkυ;0µ+λkσ;1

FRACTIONAL CALCULUS FORMULAS 835 µ+γ αα β +λkσ;1µα+β +λkσ;1. Also satisfy the following conditions: Reµσ min 1 j M Re < 1+min[ReβReα+α γreα+β γ Reηυ min 1 j M Re < 1+min[ReβReα+α γreα+β γ. and the conditions i and ii in Theorem 1 are also satisfied. Proof. Using the series definition 1.12 and replacing the H-function occurring therein by its well-known Mellin-Barnes contour integral given by 1.3 to the integrand of 3.8 and then interchanging the order of the integral sign and the summation and finally applying the 1.8 to the resulting integrals we can get the expression as in the right-hand side of 3.8. In view of the relation 2.7 then we get the following corollary concerning right-sided Saigo fractional integral operator [18 19 20. Corollary 2.1. Let αβγµηδυzab C; λσ > 0 and Reα > 0. Then the following result holds true: { I αβγ t µ1 bat η Sn [t m λ bat δ [ } H MN PQ zt σ bat υ a j α j ;A j 1N a j α j N+1P x b j 1M b j ;B j M+1Q 3.9 b η x µβ1 H 03:MN:10 33:PQ:01 n mk A nk b δk x λk [ zx σ b υ F 1 a jα j ;A j 1N a j α j N+1P F 2 b j 1M b j ;B j M+1Q 01;1 where F 1 1η δkυ;1µβ +λkσ;1 µγ +λkσ;1; and F 2 1η δkυ;0µ+λkσ;1 µ+λk αβ γσ;1. Also satisfy the following conditions: Reµσ min 1 j M Re < 1+min[ReβReγ Reηυ min 1 j M Re < 1+min[ReβReγ the conditions i and ii given in Theorem 1 are also satisfied. If we set β α in 3.9 we obtain the following result concerning rightsided Riemann-Liouville fractional integral operator [14 18: Corollary 2.2. Let αµηδυzab C; λσ > 0 and Reα > 0. Then { I α t µ1 bat η Sn [t m λ bat δ

836 DINESH KUMAR 3.10 where H MN PQ b η x µ+α1 [ } zt σ bat υ a j α j ;A j 1N a j α j N+1P x b j 1M b j ;B j M+1Q H 02:MN:10 22:PQ:01 n mk A nk b δk x λk [ zx σ b υ F 1 a j α j ;A j 1N a j α j N+1P F 2 b j 1M b j ;B j M+1Q 01;1 F 1 1η δkυ;1µ+α+λkσ;1;and F 2 1η δkυ;0µ+λkσ;1 and the existence conditions of the above corollary easily follows with the help of 3.9. Theorem 3. Let αα ββ γµηδυzab C λσ > 0 and Reγ > 0. Then we have { D αα ββ γ 0+ t µ1 bat η Sn [t m λ bat δ H MN PQ [ } zt σ bat υ a j α j ;A j 1N a j α j N+1P x b j 1M b j ;B j M+1Q b η x µ+α+α γ1 n mk A nk b δk x λk [ H 04:MN:10 zx σ b υ 3.11 44:PQ:01 where L 1 1η δkυ;11µλkσ;1 L 1 a j α j ;A j 1N a j α j N+1P L 2 b j 1M b j ;B j M+1Q 01;1 1µαα β +γ λkσ;11µα+β λkσ;1;and L 2 1η δkυ;01µαα +γ λkσ;1 1µαβ +γ λkσ;11µ+β λkσ;1. Satisfy the following conditions: Reµ+σ min 1 j M Re +max[0reαβreα+α +β γ > 0 Reη+υ min 1 j M Re +max[0reαβreα+α +β γ > 0 and the conditions i and ii as given in Theorem 1 are also satisfied. Proof. To prove the fractional differential formula 3.11 we first express the general class of polynomials occurring on its left-hand side in the series from given by 1.12 replace the H-function occurring therein by its well-known

FRACTIONAL CALCULUS FORMULAS 837 Mellin-Barnes contour integral given by 1.3 interchange the order of summations and make a little simplification. D 1 Dαα ββ γ n mk 1 0+ A nk θsz s ds 2πi L 1 b ηδkυs 1 a } η+δk+υs b t t µ+λk+σs1 3.12 x. Now interchanging the order of integrals and summations we obtain the following form after a little simplification: D 1 b η 3.13 n mk Γη+δk+υs+ξ Γη+δk+υsΓ1+ξ 2 1 A nk b δk θs zb υ s 2πi L 1 L 2 { D αα ββ γ 0+ t µ+λk+σs+ξ1 } x. Now using Lemma 3 we arrive at the following: D 1 b η x µ+α+α γ1 3.14 1 1 2πi L 2 Γ1+ξ n mk ξ Γµ+λk+σsγ+α+α +β +ξγµ+λk+σs+αβ+ξ Γµ+λk+σsγ+α+β +ξγµ+λk+σsβ+ξ a b ξdsdξ A nk b δk x λk 1 θs zb υ x σ s ds 2πi L 1 Γη+δk+υs+ξΓµ+λk+σs+ξ Γη+δk+υsΓµ+λk+σsγ+α+α +ξ by re-interpreting the Mellin-Barnes contour integral in terms of the H-function of two variables defined by 1.8 we obtain the right-hand side of 3.11 after little simplifications. This completes proof of Theorem 3. In view of the relation 2.8 then we get the following corollary concerning left-sided Saigo fractional derivative operator [18 19 20. Corollary 3.1. Let αβγµηδυzab C; λσ > 0 and Reα > 0. Then we have the following result: { D αβγ 0+ t µ1 bat η Sn [t m λ bat δ [ } H MN PQ zt σ bat υ a j α j ;A j 1N a j α j N+1P x b j 1M b j ;B j M+1Q 3.15 b η x µ+β1 H 03:MN:10 33:PQ:01 n mk A nk b δk x λk [ zx σ b υ dξ L 1a j α j ;A j 1N a j α j N+1P L 2b j 1M b j ;B j M+1Q 01;1

838 DINESH KUMAR where L 1 1ηδkυ;1 1µλkσ;1 1µλk αβ γσ;1; and L 2 1η δkυ;0 1µλk βσ;1 1µλk γσ;1. Also satisfy the following conditions: Reµ+σ min 1 j M Re > max[0reαβ γ Reη+υ min 1 j M Re > max[0reαβ γ the conditions i and ii given in Theorem 1 are also satisfied. Further if we set β α in the above result then we obtain the following result concerning left-sided Riemann-Liouville fractional derivative operator [14 18: Corollary 3.2. Let αµηδυzab C; λσ > 0 and Reα > 0. Then { D0+ α t µ1 bat η Sn [t m λ bat δ [ } H MN PQ zt σ bat υ a j α j ;A j 1N a j α j N+1P x b j 1M b j ;B j M+1Q 3.16 b η x µα1 H 02:MN:10 22:PQ:01 n mk A nk b δk x λk [ zx σ b υ L 1a j α j ;A j 1N a j α j N+1P L 2b j 1M b j ;B j M+1Q 01;1 where L 1 1η δkυ;11µλkσ;1; and L 2 1η δkυ;0 1µ+αλkσ;1 and the conditions of existence of the above corollary follow easily with the help of 3.15. Theorem 4. Let αα ββ γµηδυzab C λσ > 0 and Reγ > 0. Then we have { D αα ββ γ t µ1 bat η Sn [t m λ bat δ H MN PQ [ } zt σ bat υ a j α j ;A j 1N a j α j N+1P x b j 1M b j ;B j M+1Q b η x µ+α+α γ1 n mk A nk b δk x λk [ H 04:MN:10 zx σ b υ 3.17 44:PQ:01 where W 1 a j α j ;A j 1N a j α j N+1P W 2 b j 1M b j ;B j M+1Q 01;1 W 1 1η δkυ;1µ+λk +α+α γσ;1 µ+α +β γ +λkσ;1µβ +λkσ;1; and W 2 1η δkυ;0µ+λkσ;1µ+α+α +β γ +λkσ;1

FRACTIONAL CALCULUS FORMULAS 839 µ+α β +λkσ;1. Also satisfy the following conditions: Reµ+σ max Re Reaj1 1 j N α j < 1+min[ReβReγ αα kreγ α β Reη+υ max Re Reaj1 1 j N α j < 1+min[ReβReγ αα kreγ α β here k [Reγ + 1 and the conditions i and ii given in Theorem 1 are also satisfied. Proof. The result 3.17 can easily be obtained by following similar lines of Theorem 3 and take into account relation 2.4. In view of the relation 2.9 then we get the following corollary concerning right-sided Saigo fractional derivative operator [18 19 20. Corollary 4.1. Let αβγµηδυzab C; λσ > 0 and Reα > 0. Then the following result holds true: { D αβγ t µ1 bat η Sn [t m λ bat δ [ } H MN PQ zt σ bat υ a j α j ;A j 1N a j α j N+1P x b j 1M b j ;B j M+1Q 3.18 b η x µ+β1 H 03:MN:10 33:PQ:01 n mk A nk b δk x λk [ zx σ b υ W 1 a jα j ;A j 1N a j α j N+1P W 2 b j 1M b j ;B j M+1Q 01;1 where W 1 1ηδkυ;1µ+β +λkσ;1µαγ +λkσ;1; and W 2 1η δkυ;0µ+λkσ;1 µ+β γ +λkσ;1. Also satisfy the following conditions: Reµ+σ max 1 j N Reη+υ max 1 j N Reaj1 α j < 1+min[0[ReαReβ1Reα+γ Reaj1 α j < 1+min[0[ReαReβ1Reα+γ the conditions i and ii given in Theorem 1 are also satisfied. If we set β α in 3.19 we obtain the following result concerning rightsided Riemann-Liouville fractional derivative operator [14 18: Corollary 4.2. Let αµηδυzab C; λσ > 0 and Reα > 0. Then { D α t µ1 bat η Sn [t m λ bat δ

840 DINESH KUMAR 3.19 H MN PQ b η x µα1 [ } zt σ bat υ a j α j ;A j 1N a j α j N+1P x b j 1M b j ;B j M+1Q H 02:MN:10 22:PQ:01 n mk A nk b δk x λk [ zx σ b υ W 1 a jα j ;A j 1N a j α j N+1P W 2 b j 1M b j ;B j M+1Q 01;1 where W 1 1η δkυ;1µα+λkσ;1; and W 2 1η δkυ;0µ+λkσ;1 and the existence conditions of the above corollary easily follows with the help of 3.19. 4. Special cases TheH-functionreducestoalargenumberofspecialfunction[79101323 and so from Theorems 1-4 we can further obtain various fractional calculus results involving a number of special function. Here we provide a few special cases of our main findings see also [1 3 24 29 30. i If we set M 1 N P and Q Q+1 in Theorem 1 the H-function occurring in L.H.S. breaks up into the generalized Wright hypergeometric function Pψ Q [2 then Theorem 1 takes the following form after a little simplification: 4.1 { I αα ββ γ 0+ t µ1 bat η S m n [t λ bat δ } P ψ Q [zt σ bat υ 1a j α j ;A j 1P x 1b j ;B j 1Q b η x µαα +γ1 H 04:1P:10 44:PQ+1:01 n mk A nk b δk x λk [ zx σ b υ E 1 1a j α j ;A j 1P E 2 1b j ;B j 1Q 01;1 where E 1 and E 2 are same as given in Theorem 1. The conditions of validity of the above result easily follow from 3.1. ii If we take A j 1 j 1...N B j 1 j M + 1...Q the H-function reduces to the H-function. Then Theorem 1 takes the following form: { I αα ββ γ 0+ H MN PQ b η x µαα +γ1 [t λ bat δ t µ1 bat η Sn m [ } zt σ bat υ a j α j 1P x b j 1Q n mk A nk b δk x λk

4.2 FRACTIONAL CALCULUS FORMULAS 841 [ H 04:MN:10 zx σ b υ 44:PQ:01 E 1 a j α j 1P E 2 b j 1Q 01 where E 1 and E 2 are same as given in Theorem 1. The conditions of validity of the above result easily follow from 3.1. Remark 4.1. We can also obtain results for ordinary Bessel function of the first kind J υ z modified Bessel functions K υ z Macdonald function and Y υ z Neumann function generalized Bessel-Maitland function J µ υλ Kummer s confluent hypergeometric function φa; d; z Gauss s hypergeometric function 2F 1 ba;d;zt µ MacRobert s E-function Whittaker function by using the relation with H- function and these functions. iii If we set M 1N P 0 and Q 2 in 3.1 the H-function of one variable occurring in L.H.S. breaks up into the generalized Wright-Bessal function J ωb b z [5 then Theorem 1 takes the following form: { I αα ββ γ t µ1 bat η Sn [t m λ bat δ 4.3 0+ J ωb b [ zt σ bat υ} x b η x µαα +γ1 n mk A nk b δk x λk [ H 04:10:10 zx σ b υ 44:02:01 E 1 E 2 01bω;B01;1 where E 1 and E 2 are the same as given in 3.1. The conditions of validity of the above result easily follow from 3.1. iv If we reduce the H-function of one variable occurring in L.H.S. of 3.1 to a generalized Riemann- Zeta function [4 given by z r [ 4.4 φzlρ ρ+r l H12 22 z 01;11ρ1;l 01ρ1;l I αα ββ γ 0+ r0 then we arrive at the following result: { t µ1 bat η Sn m [ t λ bat δ φ b η x µαα +γ1 n mk A nk b δk x λk [ H 04:12:10 zx σ b υ 4.5 44:22:01 E 1 01;11ρ1;l E 2 01ρ1;l01;1 [ } zt σ bat υ l ρ x wheree 1 ande 2 arethe sameasgivenintheorem1. Theconditionsofvalidity of the above result can easily be followed directly from 3.1.

842 DINESH KUMAR v If we take M 1 and N P Q 2 in 3.1 then the H-function of one variable occurring in L.H.S. reduces into the poly-logarithm of complex order ν denoted by L ν z [22 eq. 1.12 then 3.1 takes the following form after simplification: { 4.6 I αα ββ γ 0+ [ t µ1 bat η Sn m t λ bat δ [ L ν zt σ bat υ} x b η x µαα +γ1 n mk A nk b δk x λk [ H 04:12:10 zx σ b υ 44:22:01 E 1 01;111;ν E 2 0101;ν 101;1 where E 1 and E 2 are the same as given in 3.1. The conditions of validity of the above result easily follow from Theorem 1. 5. Concluding remarks In this paper we have studied and given new unified fractional differintegral formulas involving a product of the H-function and Srivastava polynomials. The results have been given in terms of the product of one variable H-function and Srivastava polynomials in a compact and elegant form the results are obtained in terms of two variables H-function. A specimen of some of interesting special cases of main integral formula is presented briefly. Acknowledgment. I am grateful to the National Board of Higher Mathematics NBHM India for granting a Post-Doctoral Fellowship sanction no. 2/4037/2014/R&D-II/14131. References [1 D. Baleanu D. Kumar and S. D. Purohit Generalized fractional integration of the product of two H-functions and a general class of polynomials Int. J. Comp. Math. 2015 2015 10 pages. [2 R. G. Buschman and H. M. Srivastava The H-function associated with a certain class of Feynman integrals J. Phys. A. Math. Gen. 23 1990 4707 4710. [3 J. Choi and D. Kumar Certain unified fractional integrals and derivatives for a product of Aleph function and a general class of multivariable polynomials J. Inequal. Appl. 2014 2014 15 pages. [4 A. Erdélyi W. Magnus F. Oberhettinger and F. G. Tricomi Higher Transcendental Function. Vol. I McGraw-Hill New York-Toronto-London Reprinted: Krieger Melbourne-Florida 1953. [5 K. C. Gupta and R. C. Soni On a basic integral formula involving with the product of the H-function and Fox s H-function J. Raj. Acad. Phys. Sci. 4 2006 no. 3 157 164. [6 A. A. Inayat-Hussain New properties of hypergeometric series derivable from Feynman integrals. I. Transformation and reduction formulae J. Phys. A 20 1987 no. 13 4109 4117. [7 New properties of hypergeometric series derivable from Feynman integrals. II. A generalization of the H-function J. Phys. A 20 1987 no. 13 4119 4128. [8 A. A. Kilbas and M. Saigo H-Transforms theory and applications Chapman & Hall/CRC Press Boca Raton FL. 2004.

FRACTIONAL CALCULUS FORMULAS 843 [9 D. Kumar P. Agarwal and S. D. Purohit Generalized fractional integration of the H- function involving general class of polynomials Walailak Journal of Science and Technology WJST 11 2014 no. 12 1019 1030. [10 D. Kumar and J. Daiya Generalized fractional differentiation of the H-function involving general class of polynomials Int. J. Pure Appl. Sci. Technol. 16 2013 no. 2 42 53. [11 D. Kumar S. D. Purohit and J. Choi Generalized fractional integrals involving product of multivariable H-function and a general class of polynomials J. Nonlinear Sci. Appl. 9 2016 no. 1 8 21. [12 D. Kumar and R. K. Saxena Generalized fractional calculus of the M-Series involving F 3 hypergeometric function Sohag J. Math. 2 2015 no. 1 17 22. [13 A. M. Mathai R. K. Saxena and H. J. Haubold The H-Function: Theorey and Applications Springer New York 2010. [14 K. S. Miller and B. Ross An Introduction to the Fractional Calculus and Differential Equations A Wiley Interscience Publication. John Wiley and Sons Inc. New York 1993. [15 J. Ram and D. Kumar Generalized fractional integration involving Appell hypergeometric of the product of two H-functions Vijanana Prishad Anusandhan Patrika 54 2011 no. 3 33 43. [16 Generalized fractional integration of the ℵ-function J. Rajasthan Acad. Phy. Sci. 10 2011 no. 4 373 382. [17 A. K. Rathie A new generalization of generalized hypergeometric functions Matematiche Catania 52 1997 no. 2 297 310. [18 M. Saigo A remark on integral operators involving the Gauss hypergeometric functions Math. Rep. Kyushu Univ. 11 1977/78 no. 2 135 143. [19 M. Saigo and A. A. Kilbas Generalized fractional calculus of the H-function Fukuoka Univ. Science Reports 29 1999 no. 1 31 45. [20 M. Saigo and N. Maeda More generalization of fractional calculus Transform Methods and Special Functions Varna Bulgaria 1996 386 400. [21 S. G. Samko A. A. Kilbas and O. I. Marichev Fractional Integrals and Derivatives: Theory and Applications Gordon and Breach Yverdon et alibi 1993. [22 R. K. Saxena Functional relations involving generalized H-function Matematiche Catania 53 1998 no. 1 123 131. [23 R. K. Saxena J. Daiya and D. Kumar Fractional integration of the H-function and a general class of polynomials via pathway operator J. Indian Acad. Math. 35 2013 no. 2 261 274. [24 R. K. Saxena and D. Kumar Generalized fractional calculus of the Aleph-function involving a general class of polynomials Acta Math. Sci. Ser. B Engl. Ed. 35 2015 no. 5 1095 1110. [25 R. K. Saxena J. Ram and D. Kumar Generalized fractional differentiation of the Aleph-function associated with the Appell function F 3 Acta Ciencia Indica 38 2012 no. 4 781 792. [26 On the two-dimensional Saigo-Maeda fractional calculus associated with twodimensional Aleph transform Matematiche Catania 68 2013 no. 2 267 281. [27 Y. Singh and H. K. Mandia A study of H-function of two variables Int. J. of Inn. Res. in Sci. Engineering and Technology 2 2013 no. 9 4914 4921. [28 H. M. Srivastava A contour integral involving Fox s H-function Indian J. Math. 14 1972 1 6. [29 H. M. Srivastava R. K. Saxena and J. Ram Some multidimensional fractional integral operators involving a general class of polynomials J. Math. Anal. Appl. 193 1995 no. 2 373 389.

844 DINESH KUMAR [30 H. M. Srivastava and N. P. Singh The integration of certain product of the multivariable H-function with a general class of polynomials Rend. Circ. Mat. Palermo 2 32 1983 no. 2 157 187. Dinesh Kumar Department of Mathematics & Statistics Jai Narain Vyas University Jodhpur-342005 India E-mail address: dinesh dino03@yahoo.com