The extended Srivastava s triple hypergeometric functions and their integral representations

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1 Available online at J. Nonlinear Sci. Al , Research Article The extended Srivastava s trile hyergeometric functions and their integral reresentations Ayşegül Çetinkaya, M. Baki Yağbasan, İ. Onur Kıymaz Ahi Evran Univ., Det. of Mathematics, 41 Kırşehir, Turkey. Communicated by A. Atangana Abstract We introduce the extended Srivastava s trile hyergeometric functions by using an extension of beta function. Furthermore, some integral reresentations are given for these new functions. c 216 All rights reserved. Keywords: Beta function, Srivastava s trile hyergeometric functions, Aell s hyergeometric function, Exton s function. 21 MSC: 33C5, 33C6, 33C7. 1. Introduction Recently, various extensions of beta and related functions have aeared in the literature [1 3, 9 13, 17, 19]. Particularly, the following extension of beta function was introduced by Chaudhry et al. in [2] as B x, y = t x 1 1 t y 1 ex dt, 1.1 t1 t R > ; Rx >, Ry > when =. Later, by using this extension of beta function, Chaudhry et al. [3] extended the hyergeometric function as follows: B b + n, c b z n F a, b; c; z = a n Bb, c b n!, n= Corresonding author addresses: acetinkaya@ahievran.edu.tr Ayşegül Çetinkaya, mbakiyag@ahievran.edu.tr M. Baki Yağbasan, iokiymaz@ahievran.edu.tr İ. Onur Kıymaz Received

2 A. Çetinkaya, M. B. Yağbasan, İ. O. Kıymaz, J. Nonlinear Sci. Al , ; z < 1; Rc > Rb >. In [12], Özarslan et al. defined the extended Aell s hyergeometric function as F 1, a, b, c; d; x, y = B a + m + n, d a x n y m b n c m Ba, d a n! m!, 1.2 m,n= ; max{ x, y } < 1, and obtained the following integral reresentation Γd 1 F 1, a, b, c; d; x, y = t a 1 1 t d a 1 1 xt b 1 yt c ex dt, 1.3 ΓaΓd a t1 t > ; = and arg1 x < π, arg1 y < π; Rd > Ra >. Note that these extended functions are reduced to their original forms for =. 2. Extended Srivastava s trile hyergeometric functions Srivastava defined trile hyergeometric functions H A, H B and H C in [15, 16] and then many authors have studied some integral reresentations of these functions [4 8, 16]. In this aer, we introduce the extensions of Srivastava s trile hyergeometric functions as follows: α m+k β 1 m+n B β 2 + n + k, γ 2 β 2 x m y n z k H A, α, β 1, β 2 ; γ 1, γ 2 ; x, y, z := γ 1 m Bβ 2, γ 2 β 2 m! n! k!, 2.1 m,n,k= ; r < 1, s < 1, t < 1 r1 s, α + β 1 2m+n+k β 2 n+k B α + m + k, β 1 + m + n x m y n z k H B, α, β 1, β 2 ; γ 1, γ 2, γ 3 ; x, y, z:= γ 1 m γ 2 n γ 3 k Bα, β 1 m! n! k!, 2.2 m,n,k= and ; r + s + t + 2 rst < 1, β 1 m+n β 2 n+k B α + m + k, γ + n α x m y n z k H C, α, β 1, β 2 ; γ; x, y, z := γ n Bα, γ + n α m! n! k!, 2.3 m,n,k= ; r < 1, s < 1, t < 1, r + s + t 2 1 r1 s1 t < 2, where r := x, s := y, t := z. Obviously for =, these functions are reduced to the well-known Srivastava s trile hyergeometric functions H A, H B and H C, resectively. The extended Srivastava s trile hyergeometric functions defined by 2.1 and 2.3 can also be given with the following series reresentations: and H A, α, β 1, β 2 ; γ 1, γ 2 ; x, y, z = H C, α, β 1, β 2 ; γ; x, y, z = n= m= α m β 1 m γ 1 m F 1, β 2, β 1 + m, α + m; γ 2 ; y, z xm m!, 2.4 β 1 n β 2 n F 1, α, β 1 + n, β 2 + n; γ + n; x, z yn γ n n!, 2.5 where F 1, is the extended Aell s hyergeometric function given by 1.2. Throughout this aer, we assume that is any nonnegative real number.

3 A. Çetinkaya, M. B. Yağbasan, İ. O. Kıymaz, J. Nonlinear Sci. Al , Integral reresentations for H A, Theorem 3.1. The integral reresentations 3.1, of H A, hold for Rγ 2 > Rβ 2 > and the others hold for Rγ j > Rβ j >, j = 1, 2: Γγ 2 H A, α, β 1, β 2 ; γ 1, γ 2 ; x, y, z = Γβ 2 Γγ 2 β 2 t β2 1 1 t γ 2 β yt β 1 1 zt α 3.1 x 2F 1 α, β 1 ; γ 1 ; dt, t1 t 1 yt1 zt Γγ 1 Γγ 2 H A, α, β 1, β 2 ; γ 1, γ 2 ; x, y, z = Γβ 1 Γβ 2 Γγ 1 β 1 Γγ 2 β 2 β1 1 t β2 1 1 γ 1 β t γ 2 β yt α β 1 [1 yt1 zt x] α ex t1 t ddt, 3.2 Γγ 1 Γγ 2 H A, α, β 1, β 2 ; γ 1, γ 2 ; x, y, z = Γβ 1 Γβ 2 Γγ 1 β 1 Γγ 2 β 2 β 1 1 t β γ 1 β t γ 2 β yt β x zt α 1 Γγ 2 H A, α, β 1, β 2 ; γ 1, γ 2 ; x, y, z = Γβ 2 Γγ 2 β 2 2Γγ 2 H A, α, β 1, β 2 ; γ 1, γ 2 ; x, y, z = Γβ 2 Γγ 2 β 2 xyt α ex 1 yt1 x zt t1 t ddt, β α+β 1 γ y β z α x1 + ex 2 2F 1 α, β 1 ; γ 1 ; d, 1 + y1 + z Γγ 2 H A, α, β 1, β 2 ; γ 1, γ 2 ; x, y, z = Γβ 2 Γγ 2 β 2 sin 2 β cos 2 γ 2 β y sin 2 β 1 1 z sin 2 α 3.5 x sin 2 cos 2 2F 1 α, β 1 ; γ 1 ; 1 y sin 2 1 z sin 2 d, b b c β 2 a c γ 2 β 2 b a γ 2 α β 1 1 a β2 1 b γ 2 β 2 1 a c γ 2 α β 1 [σ, y] β 1 [σ, z] α 3.6 b a 2 c 2 2F 1 α, β 1 ; γ 1 ; ρ, y, zx d, a cb c ab

4 A. Çetinkaya, M. B. Yağbasan, İ. O. Kıymaz, J. Nonlinear Sci. Al , where σ, x = b a c b c ax, ρ, y, z = b a2 c 2 σ,yσ,z, c < a < b, and H A, α, β 1, β 2 ; γ 1, γ 2 ; x, y, z = Γγ 21 + λ β 2 Γβ 2 Γγ 2 β 2 where τ, x = 1 + λ 1 + λx, λ > 1. β2 1 1 γ 2 β λ γ 2 α β 1 [τ, y] β 1 [τ, z] α λ2 1 + λ ex 2 x 2F 1 α, β 1 ; γ 1 ; d, 1 + λ1 τ, yτ, z Proof. To get 3.1, it is enough to use 1.3 in 2.4. For the second integral reresentation 3.2, it is enough to use the following integral reresentation [14] 2F 1 a, b; c; x = Γc ΓbΓc b t b 1 1 t c b 1 1 xt a dt, Rc > Rb >, in 3.1. The integral reresentation 3.3 can be immediately gotten by utting [1 yt1 zt x] α = 1 yt α 1 x zt α 1 xyt α, 1 yt1 x zt in 3.2. The integral reresentations can be easily roved by directly using the transformations t = 1+, t = sin2, t = b c a 1+λ b a c and t = 1+λ in 3.1, resectively. 4. Integral reresentations for H B, Theorem 4.1. The function H B, has the following integral reresentations for min{rα, Rβ 1 } > : H B, α, β 1, β 2 ; γ 1, γ 2, γ 3 ; x, y, z = Γα + β 1 ΓαΓβ 1 t α 1 1 t β1 1 ex t1 t X 4 α + β 1, β 2 ; γ 1, γ 2, γ 3 ; xt1 t, y1 t, ztdt, 4.1 H B, α, β 1, β 2 ; γ 1, γ 2, γ 3 ; x, y, z = 2Γα + β 1 ΓαΓβ 1 sin 2 α 1 2 cos 2 β ex sin 2 cos 2 X 4 α + β 1, β 2 ; γ 1, γ 2, γ 3 ; x sin 2 cos 2, y cos 2, z sin 2 d, 4.2 H B, α, β 1, β 2 ; γ 1, γ 2, γ 3 ; x, y, z = Γα + β 1 b c α a c β 1 ΓαΓβ 1 b a α+β 1 1 b a α 1 b β1 1 c α β 1 ex σ1 σ a X 4 α + β 1, β 2 ; γ 1, γ 2, γ 3 ; xσ1 σ, y1 σ, zσd, 4.3 where σ = b c a b a c, c < a < b,

5 A. Çetinkaya, M. B. Yağbasan, İ. O. Kıymaz, J. Nonlinear Sci. Al , H B, α, β 1, β 2 ; γ 1, γ 2, γ 3 ; x, y, z = 2Γα + β 11 + λ α where σ = 1+λ sin2 1+λ sin 2, λ > 1, ΓαΓβ 1 H B, α, β 1, β 2 ; γ 1, γ 2, γ 3 ; x, y, z = 2Γα + β 1λ α sin 2 α 1 2 cos 2 β λ sin 2 α+β 1 ex σ1 σ X 4 α + β 1, β 2 ; γ 1, γ 2, γ 3 ; xσ1 σ, y1 σ, zσd, ΓαΓβ 1 sin 2 α 1 2 cos 2 β cos 2 + λ sin 2 α+β 1 ex σ1 σ X 4 α + β 1, β 2 ; γ 1, γ 2, γ 3 ; xσ1 σ, y1 σ, zσd, where σ = λ sin 2 cos 2 +λ sin 2, λ >. Here, Exton s function X 4 is defined by [18] where 2 r + s + t 2 < 1. X 4 a 1, a 2 ; c 1, c 2, c 3 ; x, y, z = m,n,k= a 1 2m+n+k a 2 n+k x m y n z k c 1 m c 2 n c 3 k m! n! k!, Proof. To obtain the first reresentation 4.1, it is enough to use 1.1 in 2.2. The other reresentations can be easily obtained by using the transformations t = sin 2, t = b c a b a c, t = 1+λ sin2 λ sin and t = 2 1+λ sin 2 cos 2 +λ sin 2 resectively. 5. Integral reresentations for H C, Theorem 5.1. The function H C, has the following integral reresentations under the assumtion Rγ > Rα > for 5.1, and the assumtion min{rα, Rβ 1, Rγ α β 1 } > for 5.2: Γγ H C, α, β 1, β 2 ; γ; x, y, z = ΓαΓγ α t α 1 1 t γ α 1 1 xt β 1 1 zt β 2 ex t1 t y1 t 2 F 1 β 1, β 2 ; γ α; dt, 1 xt1 zt Γγ H C, α, β 1, β 2 ; γ; x, y, z = ΓαΓβ 1 Γγ α β 1 Γγ H C, α, β 1, β 2 ; γ; x, y, z = ΓαΓγ α 5.1 t α 1 β1 1 1 t γ α 1 1 γ α β1 1 1 xt β 2 β xt y zt + yt + zxt 2 β 2 ex dtd, t1 t α β 1+β 2 γ 1 + x β z β y1 + ex 2F 1 β 1, β 2 ; γ α; d, 1 + x1 + z

6 A. Çetinkaya, M. B. Yağbasan, İ. O. Kıymaz, J. Nonlinear Sci. Al , Γγ H C, α, β 1, β 2 ; γ; x, y, z = ΓαΓγ α sin 2 α 1 2 cos 2 γ α x sin 2 β 1 1 z sin 2 β y cos 2 sin 2 cos 2 2F 1 β 1, β 2 ; γ α; 1 x sin 2 1 z sin 2 d, Γγ1 + λα H C, α, β 1, β 2 ; γ; x, y, z = ΓαΓγ α where τ, x = 1 + λ 1 + λx, λ > 1, Γγ H C, α, β 1, β 2 ; γ; x, y, z = ΓαΓγ α α 1 1 γ α λ γ β 1 β 2 [τ, x] β 1 [τ, z] β λ2 y1 + λ1 ex 2F 1 β 1, β 2 ; γ α; d, 1 + λ1 τ, xτ, z b b c α a c γ α b a γ β 1 β 2 1 a α 1 b γ α 1 a c γ β 1 β 2 [σ, x] β 1 [σ, z] β b a 2 c 2 2F 1 β 1, β 2 ; γ α; ρ, x, zy d, a cb c ab where σ, x = b a c b c ax, ρ, x, z = a cb ab c σ,xσ,z, c < a < b. Proof. All the integral reresentations resented here can be easily obtained as in the roof of Theorem Conclusions In this work, the extended Srivastava s trile hyergeometric functions denoted by H A,, H B, and H C, are defined by using an extension of beta function. Besides, the single series reresentations of functions H A, and H C, are given in terms of extended Aell s hyergeometric function F 1,. Finally, some integral reresentations for each of the extended Srivastava s trile hyergeometric functions are resented. The closed-form exressions of the integrals resented here, are resumably not available in the existing literature. For =, the secial cases of all reresentations given in this aer can be found in [4, 8, 16, 18]. Furthermore, a variety of different integral reresentations of these new functions can also be rovided by using the same transformations in [5 7]. Acknowledgment This aer is artly resented in 2nd and 4th International Eurasian Conference on Mathematical Sciences and Alications. This work is also suorted by Ahi Evran University PYO with roject number PYO-FEN References [1] M. Bozer, M. A. Özarslan, Notes on generalized gamma, beta and hyergeometric functions, J. Comut. Anal. Al., ,

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