CUBO, A Mathematical Journal Vol.22, N◦03, (351–359). December 2020 http://dx.doi.org/10.4067/S0719-06462020000300351 Received: 17 April, 2020 | Accepted: 05 October, 2020 The Multivariable Aleph-function involving the Generalized Mellin-Barnes Contour Integrals Abdi Oli1, Kelelaw Tilahun2, G. V. Reddy3 1,2Department of Mathematics, Wollo University, P.O. Box: 1145, Dessie, South Wollo, Amhara Region, Ethiopia. abdioli30@gmail.com, kta3151@gmail.com 3Department of Mathematics, Jigjiga University, P.O. Box: 1020, Jigjiga, Ethiopia gvreddy16673@gmail.com ABSTRACT In this paper, we have evaluated three definite integrals involving the product of two hypergeometric functions and multivariable Aleph-function. Certain special cases of the main results are also pointed out. RESUMEN En este art́ıculo, hemos evaluado tres integrales definidas que involucran el producto de dos funciones hipergeométricas y la función Aleph multivariada. También se señalan ciertos casos especiales del resultado principal. Keywords and Phrases: Hypergeometric function, Multivariable Aleph function. 2020 AMS Mathematics Subject Classification: 33C20, 33C05. c©2020 by the author. This open access article is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. http://dx.doi.org/10.4067/S0719-06462020000300351 352 A. Oli, K. Tilahun, G. V. Reddy CUBO 22, 3 (2020) 1 Introduction The Aleph-function is among very significant special functions and its closely related ones are widely used in physics and engineering. Therefore they are of high interest to physicists and engineers as well as mathematicians. In recent years, many integral formulas involving a diversity of special functions have been presented by many authors (see e.g., [3, 9, 12, 13, 14, 15, 16]). Motivated by these recent papers, three generalized integral formulae involving product of two hypergeometric functions and multivariable Aleph-function are established in the form of three theorems: For our study, we recall the following three integral formulas (see [5], p. 77, Equations (3.1), (3.2) and (3.3)): ∫ ∞ 0 [ ( αx + β x )2 + γ ]−ρ−1 dx = √ π Γ ( ρ + 1 2 ) 2α(4αβ + γ) ρ+ 1 2 Γ (ρ + 1) (1.1) ( α > 0; β ≥ 0; γ + 4αβ > 0; ℜ(ρ) + 1 2 > 0 ) . ∫ ∞ 0 1 x2 [ ( αx + β x )2 + γ ]−ρ−1 dx = √ π Γ ( ρ + 1 2 ) 2β (4αβ + γ) ρ+ 1 2 Γ (ρ + 1) (1.2) ( α ≥ 0; β > 0; γ + 4αβ > 0; ℜ(ρ) + 1 2 > 0 ) . ∫ ∞ 0 [ ( α + β x2 ) ( αx + β x )2 + γ ]−ρ−1 dx = √ π Γ ( ρ + 1 2 ) (4αβ + γ) ρ+ 1 2 Γ (ρ + 1) (1.3) ( α > 0; β ≥ 0; γ + 4αβ > 0; ℜ(ρ) + 1 2 > 0 ) . We also recall the following identity involving the hypergeometric series 2F1(.) ([8] p. 75, Theorem 1): If (1 − y)α+β−γ 2F1 (2α,2β; 2γ;y) = ∞ ∑ k=1 aky k, (1.4) then 2F1 ( a, b; c + 1 2 ; y ) 2F1 ( c − a, c − b; c + 1 2 ; y ) = ∞ ∑ k=0 (c) k ( c + 1 2 ) k aky k. (1.5) The multivariable Aleph-function defined by Sharma and Ahmad [6] as: ℵ (z1, z2, ..., zr) = ℵ0, n: m1n1; m2n2; ....;mrnr pi, qi, τi; R; p i(1) , q i(1) , τ i(1) ; R(1) , ..., p i(r) , q i(r) , τ i(r) ; R(r)      z1 ... zr ∣ ∣ ∣ ∣ ∣ ∣ ∣ B1 : B2 B3 : B4      CUBO 22, 3 (2020) The Multivariable Aleph-function involving the Generalized . . . 353 = 1 (2πω)r ∫ L1 · · · ∫ Lr ψ (ς1, · · · , ςr) r ∏ i=1 (φi (ςi) (zi) ςi)dς1 · · ·dςr (1.6) where, ω = √ −1, B1 = ( ( aj;α (1) j , · · · , α (r) j ) 1, n ) , ( τi ( aji;α (1) ji , · · · , α (r) ji ) n+1, pi ) B2 = ( ( c (1) j ,γ (r) j ) 1, n1 ) , ( τi(1) ( c (1) ji(1) ,γ (1) ji(1) ) n1+1, p i(1) ) ; · · · ; ( ( c (r) j ,γ (r) j ) 1, nr ) , ( τi(r) ( c (r) ji(r) ,γ (r) ji(r) ) nr+1, p i(r) ) B3 = ( τi ( bji;β (1) ji , · · · , β (r) ji ) m+1,qi ) B4 = ( ( d (1) j ,δ (1) j ) 1, m1 ) , ( τi(1) ( d (1) ji(1) ,δ (1) ji(1) ) m1+1, q i(1) ) ; · · · ; ( ( d (r) j ,δ (r) j ) 1, mr ) , ( τi(r) ( d (r) ji(r) ,δ (r) ji(r) ) mr+1, q i(r) ) and ψ (ς1, · · · , ςr) = ∏n j=1 Γ ( 1 − aj + ∑r k=1 α (k) j ςk ) ∑R i=1 [ τi ∏pi j=n+1 Γ ( aji − ∑r k=1 α (k) ji ςk ) ∏qi j=1 Γ ( 1 − bji + ∑r k=1 β (k) ji ςk )], (1.7) φk (ςk) = ∏mk j=1 Γ ( d (k) j − δ (k) j ςk ) ∏nk j=1 Γ ( 1 − c(k)j + γ (k) j ςk ) ∑R(k) i(k)=1 [ τi(k) ∏q i(k) j=mk+1 Γ ( 1 − d(k) ji(k) + δ (k) ji(k) ςk ) ∏p i(k) j=nk+1 Γ ( c (k) ji(k) − γ(k) ji(k) ςk )], (1.8) The parameters d (k) ji(k) (j = mk + 1, · · · , qi(k) ), (k = 1, · · · , r; i = 1, · · · , R & i(k) = 1, · · · ,R(k) are complex numbers. Also positive real numbers α’s, β’s, γ’ sand δ’s for standardiza- tion purpose such that U (k) i = n ∑ j=1 α (k) j + τi pi ∑ j=n+1 α (k) ji + nk ∑ j=1 γ (k) j + τi(k) p i(k) ∑ j=nk+1 γ (k) ji(k) − τi qi ∑ j=1 β (k) ji − mk ∑ j=1 δ (k) j − τi(k) q i(k) ∑ j=mk+1 δ (k) ji(k) ≤ 0 (1.9) 354 A. Oli, K. Tilahun, G. V. Reddy CUBO 22, 3 (2020) The real numbers τi > 0 (i = 1, ...,R) and τi(k) > 0 (i = 1, · · · , R). The contour is in the sk−plane and run from σ − ω ∞toσ + ω ∞, where σ is real number with loop, if neces- sary, ensure that the poles of Γ ( d (k) j − δ (k) j ςk ) with j = 1, . . . , mk are separated from those of Γ ( 1 − aj + ∑r k=1 α (k) j ςk ) with j = 1, . . . , n and Γ ( 1 − c(k)j + γ (k) j ςk ) with j = 1, . . . , nk to the left of the contour Lk. The condition for absolute convergence of multiple Mellin-Barnes type con- tours (1.6) can be obtained by extension of corresponding conditions for multi variable H-function as: |arg zk| < 12A (k) i π where. A (k) i = n ∑ j=1 α (k) j − τi pi ∑ j=n+1 α (k) ji − τi qi ∑ j=1 β (k) ji + nk ∑ j=1 γ (k) j − τi(k) p i(k) ∑ j=nk+1 γ (k) ji(k) + mk ∑ j=1 δ (k) j − τi(k) q i(k) ∑ j=mk+1 δ (k) ji(k) > 0 (1.10) with k = 1, · · · , r; i = 1, · · · , R and i(k) = 1, · · · , R(k). Remark 1: By setting τi = τi (k) = 1, the multivariable Aleph function reduces to multivari- able I-function (see [4, 7]). Remark 2: By setting τi = τi (k) = 1 (k = 1, ...,r) and R = R (1) =, ...,R(r) = 1, the multivariable Aleph-function reduces to multivariable H-function defined by Srivastava and Panda [10]. Remark 3: When we set r = 1, the multivariable Aleph function reduces to Aleph-function of one variable defined by Sudland [11]. 2 Main Results Theorem 2.1. Let α > 0, β ≥ 0, γ + 4αβ > 0, µi > 0, η ≥ 0, ℜ(ρ) + 12 > 0; − 1 2 < α − β − γ < 1 2 ; ℜ ( λ + µi min 1≤j≤mi { Re ( d (i) j ) δ (i) j }) > 0 (i = 1, · · · , r) , and σ = [ ( αx + β x )2 + γ ] then the following formula holds: ∫ ∞ 0 σ−ρ−12F1 ( α, β; γ + 1 2 ; σ ) 2F1 ( γ − α, γ − β; γ + 1 2 ; σ ) ×ℵ ( z1σ −η1, ..., zrσ −ηr ) dx = √ π 2α(4αβ + γ) ρ+ 1 2 ∞ ∑ h=0 1 (4αβ + γ) −h (γ) h ( γ + 1 2 ) h ah × ℵ0, n+1: m1n1; m2n2; ....;mrnr pi+1, qi+1, τi; R; p i(1) , q i(1) , τ i(1) ; R(1) , ..., p i(r) , q i(r) , τ i(r) ; R(r) CUBO 22, 3 (2020) The Multivariable Aleph-function involving the Generalized . . . 355      z1 (4αβ+γ)η1 ... zr (4αβ+γ)ηr ∣ ∣ ∣ ∣ ∣ ∣ ∣ ( −1 2 − ρ + h;η1, · · · ,ηr ) ,B1 : B2 (−ρ + h;η1, · · · ,ηr) ,B3 : B4      . (2.1) Proof. Assume that Ω in L.H.S. of (2.1), then by virtue of equation (1.5) and (1.6), we have the following Ω = ∫ ∞ 0 σ−ρ−1 ∞ ∑ h=0 (γ) h ( γ + 1 2 ) h ahσ h ℵ0, n: m1n1, m2n2, ...., mrnr pi, qi, τi; R; p i(1) , q i(1) , τ i(1) ; R(1) , ..., p i(r) , q i(r) , τ i(r) ; R(r)      z1σ −η1 ... zrσ −ηr ∣ ∣ ∣ ∣ ∣ ∣ ∣ B1 : B2 B3 : B4      dx = ∫ ∞ 0 σ −ρ−1 ∞ ∑ h=0 (γ) h ( γ + 1 2 ) h ahσ h × { 1 (2πω)r ∫ L1 · · · ∫ Lr ψ (ς1, · · · , ςr) r ∏ i=1 ( φi (ςi) ( ziσ −ηi )ςi ) dς1 · · ·dςr } dx = ∞ ∑ h=0 (γ) h ( γ + 1 2 ) h ah { 1 (2πω)r ∫ L1 · · · ∫ Lr ψ (ς1, · · · , ςr) r ∏ i=1 (φi (ςi) (zi) ςi)dς1 · · ·dςr } × ∫ ∞ 0 σ −ρ−1+h− ∑ s k=1 ηkςkdx By using equation (1.1), we can obtain the following equation Ω = ∞ ∑ h=0 (γ) h ( γ + 1 2 ) h ah { 1 (2πω)r ∫ L1 · · · ∫ Lr ψ (ς1, · · · , ςr) r ∏ i=1 (ϕi (ςi) (zi) ςi)dς1 · · ·dςr } × √ π Γ ( ρ − h + ∑s k=1 ηkςk + 1 2 ) 2α(4αβ + γ) ρ+ 1 2 −h+ ∑ s k=1 ηkςk Γ (ρ − h + ∑s k=1 ηkςk + 1) = ∞ ∑ h=0 (γ) h ah ( γ + 1 2 ) h √ π Γ ( ρ − h + ∑s k=1 ηkςk + 1 2 ) 2α(4αβ + γ) ρ+ 1 2 −h+ ∑ s k=1 ηkςk Γ (ρ − h + ∑s k=1 ηkςk + 1) × 1 (2πω)r ∫ L1 · · · ∫ Lr ψ (ς1, · · · , ςr) r ∏ i=1 (ϕi (ςi) (zi) ςi)dς1 · · ·dςr = √ π 2α(4αβ + γ) ρ+ 1 2 ∞ ∑ h=0 (γ) h ( γ + 1 2 ) h ah (4αβ + γ) −h 356 A. Oli, K. Tilahun, G. V. Reddy CUBO 22, 3 (2020) × Γ ( ρ − h + ∑s k=1 ηkςk + 1 2 ) (4αβ + γ) ∑ s k=1 ηkςk Γ (ρ − h + ∑s k=1 ηkςk + 1) × 1 (2πω)r ∫ L1 · · · ∫ Lr ψ (ς1, · · · , ςr) r ∏ i=1 (ϕi (ςi) (zi) ςi)dς1 · · ·dςr = √ π 2α(4αβ + γ) ρ+ 1 2 ∞ ∑ h=0 1 (4αβ + γ) −h (γ) h ( γ + 1 2 ) h ahx { Γ ( ρ − h + ∑s k=1 ηkςk + 1 2 ) Γ (ρ − h + ∑s k=1 ηkςk + 1) } × 1 (2πω)r ∫ L1 · · · ∫ Lr ψ (ς1, · · · , ςr) r ∏ i=1 ( φi (ςi) [ zi (4αβ + γ) ηi ]ςi ) dς1 · · ·dςr we readily arrive at the right hand side of (2.1) in view of the presentation of Aleph function in Mellin Barnes contour integral. Theorem 2.2. Let α ≥ 0, β > 0, γ + 4αβ > 0, µi > 0, η ≥ 0, ℜ(ρ)+ 12 > 0; − 1 2 < α−β −γ < 1 2 ℜ ( λ + µi min 1≤j≤mi { Re ( d (i) j ) δ (i) j }) > 0 (i = 1, · · · , r) , and σ = [ ( αx + β x )2 + γ ] then the following formula holds: ∫ ∞ 0 1 x2 σ−ρ−12F1 ( α, β; γ + 1 2 ; σ ) 2F1 ( γ − α, γ − β; γ + 1 2 ; σ ) ×ℵ ( z1σ −η1, ..., zrσ −ηr ) dx = √ π 2β (4αβ + γ) ρ+ 1 2 ∞ ∑ h=0 1 (4αβ + γ) −h (γ)h ( γ + 1 2 ) h ah × ℵ0, n+1: m1n1; m2n2; ....;mrnr pi+1, qi+1, τi; R; p i(1) , q i(1) , τ i(1) ; R(1) , ..., p i(r) , q i(r) , τ i(r) ; R(r) ×      z1 (4αβ+γ)η1 ... zr (4αβ+γ)ηr ∣ ∣ ∣ ∣ ∣ ∣ ∣ ( −1 2 − ρ + h;η1, · · · ,ηr ) ,B1 : B2 (−ρ + h;η1, · · · ,ηr) ,B3 : B4      . (2.2) Proof. In the similar manner of Theorem 2.1 and using (1.2) we easily arrive at the result (2.2). Theorem 2.3. Let α > 0, β > 0, γ +4αβ > 0, µi > 0, η ≥ 0, ℜ(ρ)+ 12 > 0; − 1 2 < α−β−γ < 1 2 ; ℜ ( λ + µi min 1≤j≤mi { Re ( d (i) j ) δ (i) j }) > 0 (i = 1, · · · , r) , and σ = [ ( αx + β x )2 + γ ] then the following formula holds: ∫ ∞ 0 ( α + β x2 ) σ−ρ−12F1 ( α, β; γ + 1 2 ; σ ) 2F1 ( γ − α, γ − β; γ + 1 2 ; σ ) ×ℵ ( z1σ −η1, ..., zrσ −ηr ) dx CUBO 22, 3 (2020) The Multivariable Aleph-function involving the Generalized . . . 357 = √ π (4αβ + γ) ρ+ 1 2 ∞ ∑ h=0 1 (4αβ + γ) −h (γ) h ( γ + 1 2 ) h ah × ℵ0, n+1: m1n1; m2n2; ....;mrnr pi+1, qi+1, τi; R; p i(1) , q i(1) , τ i(1) ; R(1) , ..., p i(r) , q i(r) , τ i(r) ; R(r)      z1 (4αβ+γ)η1 ... zr (4αβ+γ)ηr ∣ ∣ ∣ ∣ ∣ ∣ ∣ ( −1 2 − ρ + h;η1, · · · ,ηr ) ,B1 : B2 (−ρ + h;η1, · · · ,ηr) ,B3 : B4      . (2.3) Proof. In the similar way of Theorem 2.1 and using (1.3) we easily arrive at the result (2.3). 3 Special Cases (1) If we put τi = 1, in (2.1), (2.2) and (2.3), we get the results in terms of multivariable I-function [4, 7]. 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Debalkie, “A new class of integral relation involving Aleph-functions”, Surveys in Mathematics and its Applications, vol. 12, pp. 193–201, 2017. Introduction Main Results Special Cases Conclusion