Integral Transforms and Fractional Integral Operators Associated with S-Generalized Gauss Hypergeometric Function

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1 Global Journal of Pure and Applied Mathematics. ISSN Volume 13, Number 9 217, pp Research India Publications Integral Transforms and Fractional Integral Operators Associated with S-Generalized Gauss Hypergeometric Function Sanjay Bhatter and Richa Sharma 1 1 Department of Mathematics, Malaviya National Institute of Technology, Jaipur 3217, Rajasthan, India. Abstract In this paper, we first find the Euler Beta, Laplace, Whittaker, Sumudu and Hankel integral transforms of the S-Generalized Gauss hypergeometric function. Net, we obtain the image of S-generalized Gauss hypergeometric function under the certain fractional operators Saigo, Erdélyi, Kober, Riemann Liouville, Weyl fractional integral operators and Riemann Liouville fractional Derivative. AMS subject classification: 33C2, 26A33, 44A2. Keywords: S-Generalized Gauss Hypergeometric Function, Fractional Integral Operators, Integral Transform. 1. Introduction and Definitions The S-Generalized Gauss hypergeometric function: F α,β;τ,µ P a, b; c; z was introduced and investigated by Srivastava et al. [13, p.35,eq.1.12]. It is represented in the following manner: F α,β;τ,µ p a, b; c; z = B p α,β;τ,µ b + n, c b z n a n Bb,c b n! n= z < Rp ; min {Rα, Rβ, Rτ, Rµ} > ; Rc > Rb >

2 538 Sanjay Bhatter and Richa Sharma in terms of the classical Beta function Bλ, µ and the S-generalized Beta function B p α,β;τ,µ, y, which was also defined by Srivastava et al. [13, p.35, Eq. 1.13] as follows: 1 B p α,β;τ,µ, y := t 1 1 t y 1 p 1F 1 α; β; t τ 1 t µ dt 1.2 Rp ; min {R, Ry, Rα, Rβ} > ; min {Rτ, Rµ} > and λ n denotes the Pochhammer symbol defined for λ C by see [15, p. 2 and pp. 4-6]; see also [14, p. 2]: { Ɣλ + n 1, n = λ n = = 1.3 Ɣλ λλ λ + n 1, n N := {1, 2, 3, } provided that the gamma quotient eists see, for details, [17, p.16 et seq.] and [18, p.22 et seq.]. For τ = µ, the S-generalized Gauss hypergeometric function defined by 1.1 reduces to the the following generalized Gauss hypergeometric function F α,β;τ p a, b; c; zstudied earlier by parmar[1, p.44]: F α,β;τ p a, b; c; z = B p α,β;τ b + n, c b z n a n Bb,c b n! n= z < Rp ; min {Rα, Rβ, Rτ} > ; Rc > Rb > which, in the further special case when τ =1,reduces to the following etension of the generalized Gauss hypergeometric function see, e.g. [9, p.466,section 3];see also [8, p.39]: F α,β p a, b; c; z = B p α,β b + n, c b z n a n Bb,c b n! n= z < Rp : min {Rα, Rβ} > ; Rc > Rb > upon setting α = β in 1.5, we arrive at the following etended Gauss hypergeometric function see [3, p.591, Eqs.2.1 and 2.2]: F p a, b; c; z = B p b + n, c b z n a n Bb,c b n! n= z < Rp ; Rc > Rb >

3 Integral Transforms and Fractional Integral Operators Integral Transforms associated with S-Generalized Gauss Hypergeometric Function In this section, we have find the certain integral transform like that Euler, Laplace, Whittaker, Sumudu and Hankel integral transforms of the S-Generalized Gauss hypergeometric functions F p α,β;τ,µ which is defined by 1.1. Theorem 2.1. If Rp, min {Rα, Rβ, Rτ, Rµ} >, Rc > Rb > and Rl >, Rm > are parameters. Then, the following Beta transform holds: { } B F p α,β;τ,µ l + m, b; c; z; l, m = Bl, mf p α,β;τ,µ l, b; c; < where the beta transform of fzdefined as see [12] B {fz; l, m} = 1 z l 1 1 z m 1 fzdz 2.2 Further, it is assumed that the involved Euler beta transforms F p α,β;τ,µ. eist. Proof. To prove the result 2.1, by taking the Beta transform 2.2 of 1.1, we obtain 1 z l 1 1 z m 1 F p α,β;τ,µ l + m, b; c; zdz Net, we epress S-Generalized Gauss Hypergeometric Function F p α,β;τ,µ in the series form with the help of 1.1 and then changing the order of integration and summation which is permissible under the conditions stated, we get B p α,β;τ,µ b + n, c b n 1 l + m n z l+n 1 1 z m 1 dz Bb,c b n! n= Finally, with the help of Beta function and 1.1, we get the desired result 2.1 after a little simplification. Theorem 2.2. If y, Rs >, min {Rα, Rβ, Rτ, Rµ} >, Rc > Rb >, Rp and y < 1 are parameters. Then, the following Laplace s transform holds: [ ] L z l 1 F p α,β;τ,µ a, b; c; yz s = Ɣl s l 1 F p α,β;τ,µ a,b,l; c; y 2.3 s

4 54 Sanjay Bhatter and Richa Sharma where the Laplace transform of fzdefined as see [12] L[fz]s = e st fzdz Rs > 2.4 Proof. In order to prove the assertion 2.3, by taking the Laplace transform 2.4 of 1.1, we obtain e sz z l 1 F p α,β;τ,µ a, b; c; yzdz Net, we epress S-Generalized Gauss Hypergeometric Function F p α,β;τ,µ in the series form with the help of 1.1 and then changing the order of integration and summation which is permissible under the conditions stated, we get B p α,β;τ,µ b + n, c b y n a n Bb,c b n! n= e sz z n+l 1 dz Finally, with the help of Gamma function and 1.1, we get the desired result 2.3 after a little simplification. Theorem 2.3. If w/ν < 1, Rρ >, Rν >, min {Rα, Rβ, Rτ, Rµ} >, Rc > Rb > and Rp are parameters. Then, the following Whittaker transform holds: W ρ k,m [ F α,β;τ,µ p ] a, b; c; wz : ν = 1 Ɣ m + ρɣ1 2 m + ρ ν ρ Ɣ 1 2 k + ρ 2F p,1 where Whittaker transform of fzis defined as see, [6] W ρ k,m [fz: ν] = a,b, m + ρ, 1 2 m + ρ; c, 1 k + ρ; w ν 2.5 z ρ 1 e νz/2 W k,m νzf zdz 2.6 Proof. In order to prove the assertion 2.5, by taking the Whittaker transform 2.6 of 1.1, we obtain z ρ 1 e νz/2 W k,m νzf p α,β;τ,µ a, b; c; wzdz

5 Integral Transforms and Fractional Integral Operators 541 Net, we epress S-Generalized Gauss Hypergeometric Function F p α,β;τ,µ in the series form with the help of 1.1 and then changing the order of integration and summation which is permissible under the conditions stated, we get B p α,β;τ,µ b + n, c b a n Bb,c b n= Now, with the help of result [7, p. 56, Eq. 2.41], we obtain 1 ν ρ B p α,β;τ,µ b + n, c b a n Bb,c b n= w ν n n! w n n! z ρ+n 1 e νz/2 W k,m νzdz Ɣ m + n + ρɣ1 2 m + n + ρ Ɣ 1 2 k + n + ρ After a little simplification, we get the desired result 2.3. Theorem 2.4. Suppose that uy < 1, min {Rα, Rβ, Rτ, Rµ} >, Rc > Rb > and Rp are parameters. Then, the following Sumudu transform holds: [ ] S F p α,β;τ,µ α,β;τ µ a, b; c; yz; u = 1 F p a,b,1; c; uy 2.7 where the Sumudu Transform of fzdefined as see, [2] S [fz; u] = e z f uzdz [u τ 1,τ 2 ; τ 1,τ 2 > ] 2.8 Proof. In order to prove the assertion 2.7, by taking the Sumudu transform 2.8 of 1.1, we obtain e z F p α,β;τ,µ a, b; c; uyzdz Net, we epress S-Generalized Gauss Hypergeometric Function F p α,β;τ,µ in the series form with the help of 1.1 and then changing the order of integration and summation which is permissible under the conditions stated, we get B p α,β;τ,µ b + n, c b uy n a n Bb,c b n! n= e z z n dz Finally, with the help of Gamma function and 1.1, we get the desired result 2.7 after a little simplification.

6 542 Sanjay Bhatter and Richa Sharma Theorem 2.5. Suppose that Rp, min {Rα, Rβ, Rτ, Rµ} >, Rc > Rb > are parameters. Then, the following Hankel transform holds: [ ] F k F α,β;τ,µ a, b; c; ut : ν = p 2Ɣβ k 2 ƔαƔaBb, c b H,1:1,2;1,2 1,1:3,1;3,1 16u k 4 1 p 1 b; 1,τ: A 1 c; 1,τ + µ : B 2.9 where A = 1 a,1, ν 2, 1 ν, 2 2, 1 ; 1, 1, 1 c + b, µ, β, 1 2 B =, 1; α, 1 and Hankel transform defined of ftdefined as see, [5] F k [ft: ν] = tj ν ktftdt 2.1 Proof. In order to prove the assertion 2.9, by taking the Hankel transform 2.1 of 1.1, we obtain ϒ = tj ν ktf p α,β;τ,µ a, b; c; utdt Net, with the help of result [16, p.18, Eq.2.6.5], we get ϒ = F p α,β;τ,µ a, b; c; utth 1, kt,2 2 2 ν 2, 1, ν 2, 1 dt Further, with the help of result given by Bansal and Jain [2, p. 118, Eq.2.1], we get the desired result 2.9 after a little simplification. Remark 2.6. If we take τ = µ in Theorems 2.1 to 2.3 its reduces to the corresponding results due to Choi [4]. It may be remarked that the known result of Theorems 2.1 to 2.3 when τ = µ = 1 immediately reduce to the corresponding results due to Agarwal [1].

7 Integral Transforms and Fractional Integral Operators Fractional Calculus of the S-Generalized Gauss Hypergeometric Function In this section, we establish image formulas for the S-Generalized Gauss Hypergeometric Function under the Saigo fractional integral operators and also point out their special cases which are believed to be new. Theorem 3.1. If >, λ, ν, η C be parameters such that, Rρ > 1, Rρ ν +η > 1 and Rρ + λ + η > 1. Then, the following Saigo fractional integral formula holds: I λ,ν,η, t ρ 1 F p α,β;τ,µ a, b; c; t a,b,ρ ν + η, ρ; = ρ ν 1 ƔρƔρ ν + η Ɣρ νɣρ + λ + η 2 F α,β;τ,µ p,2 3.1 c, ρ ν,ρ + λ + η; where Saigo hypergeometric fractional integral operators of ftis defined as see, [7] I λ,ν,η, ft = λ ν Ɣλ t λ 1 2F 1 λ + ν, η; λ; 1 t ftdt 3.2 > and λ, ν, η C Proof. In order to prove the assertion 3.1, by taking the Saigo fractional integral 3.2 of 1.1, we obtain 1 = λ ν Ɣλ t λ 1 2F 1 λ + ν, η; λ; 1 t t ρ 1 F p α,β;τ,µ a, b; c; tdt Net, we epress S-Generalized Gauss Hypergeometric Function F p α,β;τ,µ. in the series form with the help of 1.1 and then changing the order of integration and summation which is permissible under the conditions stated, we get B p α,β;τ,µ b + r, c b λ ν 1 = a r Bb,c br! Ɣλ r= t r+ρ 1 t λ 1 2F 1 λ + ν, η; λ; 1 t dt Now, with the help of result [11], we obtain B p α,β;τ,µ b + r, c b Ɣr + ρɣρ + r + η ν 1 = a r Bb,c br! Ɣρ + r νɣρ + r + λ + η r+ρ ν 1 r=

8 544 Sanjay Bhatter and Richa Sharma Finally, with the help of 1.1, we get the desired result 3.1 after a little simplification. Theorem 3.2. If >, λ, ν, η C be parameters such that, Rρ > 1, Rρ + ν > 1 and Rλ + ν + ρ + η > 1. Then, the following Saigo fractional integral formula holds: J, λ,ν,η t ρ F p α,β;τ,µ a,b; c; 1 Ɣν + ρɣη + ρ = t ƔρƔλ + ν + η + ρ ρ ν a,b,ν + ρ,η + ρ; 2F α,β;τ,µ 1 p,2 c, ρ, ν + λ + η + ρ; 3.3 where Saigo hypergeometric fractional integral operators of ftis defined as see, [7] J λ,ν,η, ft = 1 t λ 1 t λ ν 2F 1 λ + ν, η; λ; 1 ftdt Ɣλ t 3.4 > and λ, ν, η C Proof. In order to prove the assertion 3.3, by taking the Saigo fractional integral 3.4 of 1.1, we obtain 1 = 1 t λ 1 t λ ν 2F 1 λ + ν, η; λ; 1 t ρ F p α,β;τ,µ a,b; c; 1 dt Ɣλ t t Net, we epress S-Generalized Gauss Hypergeometric Function F p α,β;τ,µ in the series form with the help of 1.1 and then changing the order of integration and summation which is permissible under the conditions stated, we get B p α,β;τ,µ b + r, c b 1 1 = a r Bb,c br! Ɣλ r= t λ 1 t ρ λ ν r 2F 1 µ + ν, η; µ; 1 t Now, with the help of result [11], we obtain 1 = dt B p α,β;τ,µ b + r, c b Ɣν + r + ρɣη + r + ρ a r Bb,c br! Ɣr + ρɣλ + ν + η + r + ρ r ρ ν r= Finally, with the help of 1.1, we get the desired result 3.3 after a little simplification.

9 Integral Transforms and Fractional Integral Operators 545 Corollary 3.3. If we put ν = in 3.1, then Saigo hypergeometric fractional integral operator reduces to Erdélyi fractional integral operator of S-Generalized Gauss Hypergeometric Function E λ,η, tρ 1 F p α,β;τ,µ a, b; c; t = ρ 1 Ɣρ + η Ɣρ + λ + η 1 F α,β;τ,µ p,1 a,b,ρ + η; c, ρ + λ + η; 3.5 where Erdélyi fractional integral operator of ftis defined as see, [19] E λ,η, ft = λ η t λ 1 t η ftdt Rλ >, Rη > Ɣλ 3.6 provided that the conditions are easily obtainable from the eisting conditions of 3.1 are satisfied. Corollary 3.4. If we put ν = in 3.3, then Saigo hypergeometric fractional integral operator reduces to Kober fractional integral operator of S-Generalized Gauss Hypergeometric Function K, λ,η t ρ F p α,β;τ,µ a,b; c; 1 t a,b,η + ρ; Ɣη + ρ = Ɣλ + η + ρ ρ 1F α,β;τ,µ 1 p,1 3.7 c, λ + η + ρ; where Kober fractional integral operator of ftis defined as see, [19] K λ,η, ft = η Ɣλ t λ 1 t λ η ftdt Rλ >, Rη > 3.8 provided that the conditions are easily obtainable from the eisting conditions of 3.3 are satisfied. Corollary 3.5. If we put ν = λin 3.1, then Saigo hypergeometric fractional integral operator reduces to Riemann-liouville fractional integral operator of S-Generalized Gauss Hypergeometric Function R, λ tρ 1 F p α,β;τ,µ a, b; c; t = ρ+λ 1 Ɣρ Ɣρ + λ 1 F α,β;τ,µ p,1 a,b,ρ; c, ρ + λ + η; 3.9

10 546 Sanjay Bhatter and Richa Sharma where Riemann Liouville Fractional integral Operator of ftis defined as see, [19] R λ, ft = 1 Ɣλ t λ 1 ftdt Rλ > 3.1 Corollary 3.6. If we put ν = λ in 3.3, then Saigo hypergeometric fractional integral operator reduces to Weyl type fractional integral operator of S-Generalized Gauss Hypergeometric Function W, λ t ρ F p α,β;τ,µ a,b; c; 1 = t Ɣρ λ Ɣρ λ ρ 1F α,β;τ,µ p,1 where Weyl Fractional integral Operator of ftis defined as see, [19] W µ, ft = 1 Ɣµ a,b,ρ λ; 1 c, ρ; 3.11 t µ 1 ftdt Rµ > 3.12 Corollary 3.7. If we replace λ by λ in 3.9, then Riemann-Liouville Fractional Integral Operator reduces to Riemann-Liouville fractional derivative operator of S- Generalized Gauss Hypergeometric Function D, λ tρ 1 F p α,β;τ,µ a, b; c; t = ρ λ 1 Ɣρ Ɣρ λ 1 F α,β;τ,µ p,1 References a,b,ρ; c, ρ λ + η; 3.13 [1] Agarwal P., Certain properties of the Generalized Gauss Hypergeometric Functions, Applied Mathematics & Information Sciences, 85, , 214. [2] Bansal M. K. and Jain R., Certain New Results of The S-Generalized Gauss Hypergeometric Function Transform, South East Asian J. of Math.& Math. Sci., 122, , 216. [3] Chaudhry M. A., Qadir A., Srivastava H. M. and Paris R. B., Etended Hypergeometric and Confluent Hypergeometric Functions, Appl. Math. Comput. 159, , 24. [4] Choi J. and Agarwal P., Certain Integral Transform and Fractional Integral Formulas for the Generalized Gauss Hypergeometric Functions, Abstract and Applied Analysis, 214, 1 7, 214. [5] Debnath L. and Bhatta D., Integral Transforms and Their Applications, Chapman & Hall/CRC, Boca Raton, London, New York, 27.

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