Palestine Journal of Mathematics Vol. 8(2)(2019) , 146–158 © Palestine Polytechnic University-PPU 2019 SOME NEW RESULTS FOR HUMBERT’S DOUBLE HYPERGEOMETRIC SERIES 2 AND φ2 Ghazala Yasmin and Abdulghani Muhyi Communicated by P.K. Banerji MSC 2010 Classifications: Primary 33B15, 33C20; Secondary 33C70. Keywords and phrases: Gamma Function, Generalized Hypergeometric Function, Kummer’s Theorem, Generalized (Wright) Hypergeometric Function, Humbert’s Functions, Lavoie-Trottier Integral Formula. Abstract. This article aims to establish new expressions of Humbert’s functions 2 and φ2 with the help of the generalization of Kummer’s summation theorem. Some special cases are also obtained. Further, we present some interesting integrals involving the Humbert’s functions, which are expressed in terms of generalized (Wright) hypergeometric function. Also, some special cases were considered as an application of the presented integral formulas. 1 Introduction The theory of hypergeometric functions has a vast popularity and great interest. The usefulness of this functions lies in the fact that the solutions of many applied problems involving partial dif- ferential equations are obtainable with the help of such hypergeometric functions. For instance, the energy absorbed by some non ferromagnetic conductor sphere contained in an internal mag- netic field can be calculated with the help of such functions [12, 15]. Multi-variable hyper- geometric functions are used in physical and quantum chemical applications as well [14, 19]. Especially, many problems in gas dynamics lead to those of degenerate second order partial differential equations, which are then solvable in terms of multiple hypergeometric functions. Further, in the investigation of the boundary value problems for these partial differential equa- tions, we need decompositions for multivariate hypergeometric functions in terms of simpler hypergeometric functions of the Gauss and Humbert types. The generalized hypergeometric function (GHF) is defined by [16, p.73 (2)] (see also [1]) : 2 3 α1; α2; ::; αp; 6 7 pFq 4 z5 = pFq[α1; α2; ::; αp; β1; β2; :::; βq; z] β1; β2; :::; βq; p Q 1 (αi)m m X i=1 z = 1 + q ; (βj > 0; j = 0; 1; 2; :::; q); (1.1) Q m! m=1 (βj)m j=1 where (λ)m denotes the Pochhammer symbol (or the shifted factorial, since (1)m = m!) defined, for any complex number α, by ( α(α + 1):::(α + m − 1); m 2 N = 1; 2; 3; ::: (λ)m := 1; m = 0 (1.2) G(λ + m) = ; (λ 2 n −) G(λ) C Z0 where G(r) is the well known gamma function [16, 20] defined by Z 1 G(r) = e−xxr−1dx; (<(r) > 0): (1.3) 0 SOME NEW RESULTS FOR HUMBERT’S SERIES 2 AND φ2 147 The Beta function B(υ; τ), υ, τ 2 C, is defined by [16, 20]: (R 1 υ−1 τ−1 0 x (1 − x) dx; <(υ) > 0; <(τ) > 0 B(υ; τ) = G(υ)G(τ) : (1.4) G(υ+τ) ; <(υ) < 0; <(τ) < 0; υ; τ =6 −1; −2; −3; :::: The Wright hypergeometric function pYq is defined by [13, 16] p 2 3 Q (α1; η1); (α2; η2); ::; (αp; ηp); 1 G(αi + ηim) m X i z Y 6 z7 = =1 ; (1.5) p q 4 5 q m! m=1 Q (β1; ξ1); (β2; ξ2); :::; (βq; ξq); G(βj + ξjm) j=1 where the coefficient η1, η2,..., ηp and ξ1, ξ2, ..., ξq are positive real numbers such that q p X X (a) 1 + ξj − ηi > 0 and 0 < jzj < 1; z =6 0; j=1 i=1 q p X X −η1 −ηp ξ1 ξq (b) 1 + ξj − ηi = 0 and 0 < jzj < η1 ...ηp ξ1 ...ξq : j=1 i=1 The Humbert’s functions are defined by [2] (see also[3, 13, 20, 21]) : 1 1 s r X X (a1)s+r y z 2(a1; a2; a3; y; z) = (1.6) (a2)s(a3)r s! r! s=1 r=1 and 1 1 s r X X (a1)s(a2)s y z φ2(a1; a2; a3; y; z) = ; (1.7) (a3)s+r s! r! s=1 r=1 where the double series (1.6)-(1.7) converge absolutely at any y; z 2 C. Further, we recall some relations between hypergeometric series [3] (see also [13, 20]) and Humbert’s functions as : 2 3 − ; 6 7 2(a1; a1; a1; y; z) = exp(y + z)0F14 yz5; (1.8) a1 ; 2 1 1 1 3 2 a1; 2 a1 + 2 ; 6 27 2(a1; a3; a3; −y; y) = 2F34 − y 5; (1.9) 1 1 1 a3; 2 a3; 2 a3 + 2 ; 2 1 1 1 1 1 3 a1; 2 a2 + 2 a3 − 2 ; 2 a2 + 2 a3; 6 7 2(a1; a2; a3; y; y) = 3F34 4y5; (1.10) a2; a3; a2 + a3 − 1; 2 3 a1 ; 6 7 φ2(a1; a2 − a1; a2; y; z) = exp(z)1F14 y − z5; (1.11) a2 ; 2 0 3 a1 + a1; 0 6 7 φ2(a1; a1; a2; y; y) = 0F14 y5; (1.12) a2 ; 148 Ghazala Yasmin and Abdulghani Muhyi 2 3 a ; 1 y2 6 7 φ2(a1; a1; a3; −y; y) = 1F24 5: (1.13) 1 1 1 4 2 a3; 2 a3 + 2 ; The two identities (1.6) and (1.7) may be written as [13, 17] 2 3 1 −s; −s − a2 + 1; s X (a1)s z y 2(a1; a2; a3; y; z) = 2F16 7 ; (1.14) (a2)s 4 y 5 s! s=1 a3; where a2; a3 =6 0; −1; −2; ::: and jyj= 6 0. 2 3 1 − r; a2; r X (a1)r z y φ2(a1; a2; a3; y; z) = 2F16 7 ; (1.15) (a3)r 4 y 5 r! r=1 1 − a1 − r; where a3 =6 0; −1; −2; ::: and jyj= 6 0. In the present investigation, the results are obtained with the help of the following general- ization of the classical Kummer’s theorem [10] (see also [4]): 2 3 a1; a2; 1 G( 2 )G(1 − a2)G(1 + a1 − a2 + j) 2F16 − 17 = 4 5 a1 1 2 G(1 − a2 + 2 (j + jjj)) 1 + a1 − a2 + j; µ ν × j + j ; 1 1 1 1 1 j+1 1 1 1 1 1 j G( 2 a1 − a2 + 2 j + 1)G( 2 a1 + 2 j + 2 − [ 2 ]) G( 2 a1 − a2 + 2 j + 2 )G( 2 a1 + 2 j − [ 2 ]) (1.16) where j = 0; ±1; ±2; ±3; ±4; ±5. Here [z] denotes the greatest integer less than or equal to z. For j = 0 the identity (1.16) reduces to the classical Kummer’s theorem [16]: 2 3 a1; a2; −a G( 1 )G(a −a +1) 2 1 2 1 2 F 6 − 17 = ; (<(a ) < 1; a − a 2 n −): 2 14 5 1 1 1 2 1 2 C Z G( 2 a1 + 2 )G( 2 a1 − a2 + 1) 1 + a1 − a2; (1.17) Consider the well known identity: 1 1 1 X X X '(m) = '(2m) + '(2m + 1): (1.18) m=0 m=0 m=0 Also, we note the following identities[16, 20]: 2m α α + 1 (α)2m = 2 ; (1.19) 2 m 2 m (α)s+r = (α)s(α + s)r; (1.20) (−1)s (α)−s = : (1.21) (1 − α)s SOME NEW RESULTS FOR HUMBERT’S SERIES 2 AND φ2 149 Further, we recall the well known result of Lavoie and Trottier [11]: Z 1 x2α−1 xβ−1 22α G(α)G(β) xα−1 (1 − x)2β−1 1 − 1 − dx = ; (1.22) 0 3 4 3 G(α + β) where (<(α) > 0 and <(β) > 0) . Very recently, Choi et al. [4,6,8,9, 17, 18] have considered the reducibility of some exten- sively generalized special functions and established many results using generalized Kummer’s summation theorem in various ways. In this sequel, using the same technique, we propose to derive certain explicit expressions for Humbert’s functions 2 and φ2. Some special cases of the main results are also derived. Some integral formulas involving Humbert’s functions 2 and φ2, expanded in terms of the Wright hypergeometric function are also established. 2 Main Results We establish two generalized formulas for the Humbert’s functions 2 and φ2 asserted by the following two theorems. Theorem 2.1. For a3 =6 0; −1; −2; :::, the following identity for 2 holds true: 1 2s X (a1)2s y 0 0 2(a1; a3; a3 + j; −y; y) = (αj A2s + βj B2s) (a3)2s(2s)! s=1 1 2s+1 X (a1)2s+1 y 00 00 − (αj A2s+1 + βj B2s+1); (2.1) (a3)2s+1(2s + 1)! s=1 where 22s G( 1 ) G(2s + a ) G(a + j) A = 2 3 3 ; 2s 1 j j 1 j+1 G(2s + a3 + 2 (j + jjj)) G(s + a3 + 2 ) G(−s + 2 + 2 − [ 2 ]) 22s G( 1 ) G(2s + a ) G(a + j) B 2 3 3 ; 2s = 1 j 1 j j G(2s + a3 + (j + jjj)) G(s + a3 + − ) G(−s + − [ ]) 2 2 2 2 2 (2.2) 22s+1 G( 1 ) G(2s + a + 1) G(a + j) A = 2 3 3 ; 2s+1 1 j 1 j j+1 G(2s + a3 + 1 + 2 (j + jjj)) G(s + a3 + 2 + 2 ) G(−s + 2 − [ 2 ]) 22s+1 G( 1 ) G(2s + a + 1) G(a + j) B 2 3 3 2s+1 = 1 j j 1 j G(2s + a3 + 1 + 2 (j + jjj)) G(s + a3 + 2 ) G(−s + 2 − 2 − [ 2 ]) 0 0 for j = 0; ±1; ±2; ±3; ±4; ±5.