Some New Results for Humbert's Double

Palestine Journal of Mathematics

Vol. 8(2)(2019) , 146–158 © Palestine Polytechnic University-PPU 2019

SOME NEW RESULTS FOR HUMBERT’S DOUBLE HYPERGEOMETRIC SERIES ψ2 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 ψ2 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 differential 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 hypergeometric 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 equations, 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]) :   α1, α2, .., αp;   pFq  z = pFq[α1, α2, .., αp; β1, β2, ..., βq; z] β1, β2, ..., βq; p Q ∞ (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 ∈ N = 1, 2, 3, ... (λ)m := 1, m = 0 (1.2) Γ(λ + m) = , (λ ∈ \ −) Γ(λ) C Z0 where Γ(r) is the well known gamma function [16, 20] defined by Z ∞ Γ(r) = e−xxr−1dx, (<(r) > 0). (1.3) 0 SOME NEW RESULTS FOR HUMBERT’S SERIES ψ2 AND φ2 147

The Beta function B(υ, τ), υ, τ ∈ C, is deﬁned by [16, 20]:

(R 1 υ−1 τ−1 0 x (1 − x) dx, <(υ) > 0, <(τ) > 0 B(υ, τ) = Γ(υ)Γ(τ) . (1.4) Γ(υ+τ) , <(υ) < 0, <(τ) < 0, υ, τ =6 −1, −2, −3, ....

The Wright hypergeometric function pΨq is deﬁned by [13, 16]

p   Q (α1, η1), (α2, η2), .., (αp, ηp); ∞ Γ(αi + ηim) m X i z Ψ  z = =1 , (1.5) p q   q m! m=1 Q (β1, ξ1), (β2, ξ2), ..., (βq, ξq); Γ(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 < |z| < ∞; z =6 0, j=1 i=1 q p X X −η1 −ηp ξ1 ξq (b) 1 + ξj − ηi = 0 and 0 < |z| < η1 ...ηp ξ1 ...ξq . j=1 i=1

The Humbert’s functions are deﬁned by [2] (see also[3, 13, 20, 21]) :

∞ ∞ s r X X (a1)s+r y z ψ2(a1; a2, a3; y, z) = (1.6) (a2)s(a3)r s! r! s=1 r=1 and ∞ ∞ 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 ∈ C. Further, we recall some relations between hypergeometric series [3] (see also [13, 20]) and Humbert’s functions as :   − ;   ψ2(a1; a1, a1; y, z) = exp(y + z)0F1 yz, (1.8) a1 ;

 1 1 1  2 a1, 2 a1 + 2 ;  2 ψ2(a1; a3, a3; −y, y) = 2F3 − y , (1.9) 1 1 1 a3, 2 a3, 2 a3 + 2 ;

 1 1 1 1 1  a1, 2 a2 + 2 a3 − 2 , 2 a2 + 2 a3;   ψ2(a1; a2, a3; y, y) = 3F3 4y, (1.10) a2, a3, a2 + a3 − 1;

  a1 ;   φ2(a1, a2 − a1; a2; y, z) = exp(z)1F1 y − z, (1.11) a2 ;

 0  a1 + a1; 0   φ2(a1, a1; a2; y, y) = 0F1 y, (1.12) a2 ; 148 Ghazala Yasmin and Abdulghani Muhyi

  a ; 1 y2   φ2(a1, a1; a3; −y, y) = 1F2 . (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]   ∞ −s, −s − a2 + 1; s X (a1)s z y ψ2(a1; a2, a3; y, z) = 2F1  , (1.14) (a2)s  y  s! s=1 a3; where a2, a3 =6 0, −1, −2, ... and |y|= 6 0.   ∞ − r, a2; r X (a1)r z y φ2(a1, a2; a3; y, z) = 2F1  , (1.15) (a3)r  y  r! r=1 1 − a1 − r; where a3 =6 0, −1, −2, ... and |y|= 6 0.

In the present investigation, the results are obtained with the help of the following generalization of the classical Kummer’s theorem [10] (see also [4]):

  a1, a2; 1 Γ( 2 )Γ(1 − a2)Γ(1 + a1 − a2 + j) 2F1 − 1 =   a1 1 2 Γ(1 − a2 + 2 (j + |j|)) 1 + a1 − a2 + j;

µ ν × j + j , 1 1 1 1 1 j+1 1 1 1 1 1 j Γ( 2 a1 − a2 + 2 j + 1)Γ( 2 a1 + 2 j + 2 − [ 2 ]) Γ( 2 a1 − a2 + 2 j + 2 )Γ( 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]:   a1, a2; −a Γ( 1 )Γ(a −a +1) 2 1 2 1 2 F  − 1 = , (<(a ) < 1; a − a ∈ \ −). 2 1  1 1 1 2 1 2 C Z Γ( 2 a1 + 2 )Γ( 2 a1 − a2 + 1) 1 + a1 − a2; (1.17)

Consider the well known identity:

∞ ∞ ∞ X X X ϕ(m) = ϕ(2m) + ϕ(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 ψ2 AND φ2 149

Further, we recall the well known result of Lavoie and Trottier [11]:

Z 1 x2α−1 xβ−1 22α Γ(α)Γ(β) xα−1 (1 − x)2β−1 1 − 1 − dx = , (1.22) 0 3 4 3 Γ(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 ψ2 and φ2. Some special cases of the main results are also derived. Some integral formulas involving Humbert’s functions ψ2 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 ψ2 and φ2 asserted by the following two theorems.

Theorem 2.1. For a3 =6 0, −1, −2, ..., the following identity for ψ2 holds true:

∞ 2s X (a1)2s y 0 0 ψ2(a1; a3, a3 + j; −y, y) = (αj A2s + βj B2s) (a3)2s(2s)! s=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 Γ( 1 ) Γ(2s + a ) Γ(a + j) A = 2 3 3 , 2s 1 j j 1 j+1 Γ(2s + a3 + 2 (j + |j|)) Γ(s + a3 + 2 ) Γ(−s + 2 + 2 − [ 2 ]) 22s Γ( 1 ) Γ(2s + a ) Γ(a + j) B 2 3 3 , 2s = 1 j 1 j j Γ(2s + a3 + (j + |j|)) Γ(s + a3 + − ) Γ(−s + − [ ]) 2 2 2 2 2 (2.2) 22s+1 Γ( 1 ) Γ(2s + a + 1) Γ(a + j) A = 2 3 3 , 2s+1 1 j 1 j j+1 Γ(2s + a3 + 1 + 2 (j + |j|)) Γ(s + a3 + 2 + 2 ) Γ(−s + 2 − [ 2 ]) 22s+1 Γ( 1 ) Γ(2s + a + 1) Γ(a + j) B 2 3 3 2s+1 = 1 j j 1 j Γ(2s + a3 + 1 + 2 (j + |j|)) Γ(s + a3 + 2 ) Γ(−s + 2 − 2 − [ 2 ])

0 0 for j = 0, ±1, ±2, ±3, ±4, ±5. The coefﬁcients αj and βj can be obtained from the tables given by J. Choi [4], by substituting a and b with −2s and −2s − a3 + 1, respectively, while the 00 00 coefﬁcients αj and βj can be obtained from the same table by substituting a and b with −2s − 1 0 0 00 00 and −2s − a3, respectively. The obtained values of αj, βj, αj and βj are given in Table 1.

0 00 0 00 Table 1. Table for αj, αj , βj and βj 0 0 00 00 j αj βj αj βj 4s2 − 5a2 + 2s − −4s2 + a2 − 14s − 4s2 − 5a2 + 6s − −4s2 +a2 −8sa − 5 3 3 3 3 3 15a3 − 12 8sa3 + a3 15a3 − 10 18s − 3a3 − 8 2 2 2 2 −4s +a3 −4sa3 − −4s +a3 −4sa3 − 4 −4s − 4 −4a3 − 4 6s + a3 10s − a3 − 4

3 −2s − 3a3 − 2 −2s + a3 −2s − 3a3 − 3 −2s + a3 − 1

2 a3 −2 a3 −2 150 Ghazala Yasmin and Abdulghani Muhyi

1 −1 1 −1 1 0 1 0 1 0 -1 1 1 1 1

-2 a3 − 2 2 −2 2

-3 2s + 3a3 − 7 −2s + a3 − 3 2s + 3a3 − 6 −2s + a3 − 4 2 2 2 −4s +a3 −4sa3 + −4s2 +a3 −4sa3 + -4 4a3 − 12 4a3 − 12 10s − 7a3 + 12 6s − 9a3 + 16 −4s2 + 5a − 2s − −4s2 +a2 −8sa + −4s2 + 5a2 − 6s − −4s2 − 8sa + -5 3 3 3 3 3 35a3 + 62 26s − 9a3 + 20 35a3 + 60 22s − 13a3 + 32

Proof. In order to derive our result we ﬁrst replace y by −y and set z = y in (1.14). Further, replacing a2 and a3 by a3 and a3 + j respectively in the resultant equation, we get   ∞ −s, −s − a3 + 1; s X (a1)s (−y) ψ2(a1; a3, a3 + j; −y, y) = 2F1 − 1 . (2.3) (a3)s   s! s=1 a3 + j; Now, in view of (1.18), we separate the r.h.s of (2.3) into even and odd terms as:   ∞ 2s −2s, −2s − a3 + 1; X (a1)2sy ψ2(a1; a3, a3 + j; −y, y) = 2F1 − 1 (a3)2s(2s)!   s=1 a3 + j;   ∞ 2s+1 −2s − 1, −2s − a3; X (a1)2s+1y − 2F1 − 1. (2.4) (a3)2s+1(2s + 1)!   s=1 a3 + j;

Finally, evaluating the both resultant 2F1 with the help of the generalized Kummer’s theorm (1.16) and then, after some simpliﬁcation by using the identities (1.19)-(1.21), we arrive at the right-hand side of our general formula (2.1). This completes the proof of Theorem 2.1.

Theorem 2.2. For a3 =6 0, −1, −2, ..., the following identity for φ2 holds true:

∞ 2r X (a1)2r y φ2(a1, a1 + j; a3; −y, y) = (γj C2r + δj D2r) (a3)2r(2r)! r=1 ∞ 2r+1 X (a1)2r+1 y 0 0 − (γj C2r+1 + δj D2r+1), (2.5) (a3)2r+1(2r + 1)! r=1 where 22r Γ( 1 ) Γ(1 − a − j) Γ(1 − a − 2r) C = 2 1 1 , 2r 1 j j 1 j+1 Γ(1 − a1 + 2 (|j| − j)) Γ(−r − a1 − 2 + 1) Γ(−r + 2 + 2 − [ 2 ]) 22r Γ( 1 ) Γ(1 − a − j) Γ(1 − a − 2r) D 2 1 1 , 2r = 1 j 1 j j Γ(1 − a1 + (|j| − j)) Γ(−r − a1 − + ) Γ(−r + − [ ]) 2 2 2 2 2 (2.6) 22r+1 Γ( 1 ) Γ(1 − a − j) Γ(−a − 2r) C = 2 1 1 , 2r+1 1 j 1 j j+1 Γ(1 − a1 + 2 (|j| − j)) Γ(−r − a1 − 2 + 2 ) Γ(−r + 2 − [ 2 ]) 22r+1 Γ( 1 ) Γ(1 − a − j) Γ(−a − 2r) D 2 1 1 2r+1 = 1 j j 1 j Γ(1 − a1 + 2 (|j| − j)) Γ(−r − a1 − 2 ) Γ(−r + 2 − 2 − [ 2 ])

for j = 0, ±1, ±2, ±3, ±4, ±5. The coefﬁcients γj and δj can be obtained from the tables given by J. Choi [4], by substituting a and b with −2r and a1 + j, respectively, while the coefﬁ- 0 0 cients γj and δj can be obtained from the same table by substituting a and b with −2r − 1 and 0 0 a1 + j, respectively. The obtained values of γj, δj, γj and δj are given in Table 2. SOME NEW RESULTS FOR HUMBERT’S SERIES ψ2 AND φ2 151

0 0 Table 2. Table for γj, δj, γj and δj 0 0 j γj δj γj δj 2 2 2 2 2 2 2 2 −16r − 5a1 − 16r + a1 + −16r − 5a1 − 16r + a1 + 5 20ra1 − 48r − 12ra1 + 32r + 20ra1 − 64r − 12ra1 + 48r + 25a1 − 32 7a1 + 12 35a1 − 60 13a1 + 32 2 2 2 2 8r + a1 + 8ra1 + 8r + a1 + 8ra1 + 4 4a1 + 8r + 8 8r + 4a1 + 12 16r + 5a1 + 6 24r + 9a1 + 16

3 3a1 + 4r + 4 −4r − a1 − 2 4r + 3a1 + 6 −4r − a1 − 4

2 −2r − a1 − 1 −2 −2r − a1 − 2 −2 1 −1 1 −1 1 0 1 0 1 0 -1 1 1 1 1

-2 −2r − a1 + 1 2 −2r − a1 2

-3 −4r − 3a1 + 5 −4r − a1 + 1 −4r − 3a1 + 3 −4r − a1 − 1 2 2 2 2 8r + r + 8ra1 − 8r + a1 + 8ra1 − -4 −8r − 4a1 + 8 −8r − 4a1 + 4 16r − 3a1 + 2 8r + a1 − 4 2 2 2 2 2 2 2 2 16r + 5r + 16r + a1 + 16r + 5a1 − 16r + a1 + -5 20ra1 − 52r − 12ra1 − 28r − 20ra1 − 36r − 12ra1 − 12r + 25a1 + 32 3a1 + 2 15a1 + 10 3a1 − 8

Proof. Making use of (1.15) and using a similar argument as in the proof of the Theorem 2.1, we can establish the general formula (2.5).

Several integral representations of the Humbert’s function φ2 is derived in [5], one of which is given as follows: (γ) Z 1 Z 1 Γ xξ+y(1−ξ)η ε1−1 β2−1 φ2(β1, β2; γ; x, y) = e ξ η Γ(ε1)Γ(β2)Γ(γ − ε1 − β2) 0 0

γ−ε1−1 γ−ε1−β2−1 ×(1 − ξ) (1 − η) 1F1(β1 − ε1; γ − ε1 − β2; x(1 − ξ)(1 − η)dξdη, (2.7)

(<(γ − ε1 − β2) > 0, <(ε1) > 0, <(β2) > 0). Now, by using (2.7) and (2.5), we establish general integral relation which includes eleven identities (i.e, for j = 0, ±1, ±2, ±3, ±4, ±5) asserted by the following corollary. Corollary 2.3. The following integral relation holds true:

Z 1 Z 1 e−yξ+y(1−ξ)ηξε1−1ηa1+j−1(1 − ξ)a3−ε1−1(1 − η)a3−ε1−a1−j−1 0 0

1F1(a1 − ε1; a3 − ε1 − a1 − j; −y(1 − ξ)(1 − η)dξdη

( ∞ 2r Γ(ε1)Γ(a1 + j)Γ(γ − ε1 − a1 − j) X (a1)2r y = (γj C2r + δj D2r) Γ(a3) (a3)2r(2r)! r=1 ∞ 2r+1 ) X (a1)2r+1 y 0 0 − (γj C2r+1 + δj D2r+1) , (2.8) (a3)2r+1(2r + 1)! r=1 where (<(a3 − ε1 − a1 − j) > 0, <(ε1) > 0, <(a1 + j) > 0) and j = 0, ±1, ±2, ±3, ±4, ±5, the 0 0 coefﬁcients γj, δj, γj, δj and other notations are the same as in (2.5,2.6).

Proof. We can derive our assertion in a straightforward way. Indeed, if we set β1 = a1, β2 = a1 + j and γ = a3 in (2.7), we obtain Z 1 Z 1 Γ(a3) −yξ+y(1−ξ)η ε1−1 a1+j−1 φ2(a1, a1 + j; a3; x, y) = e ξ η Γ(ε1)Γ(a1 + j)Γ(γ − ε1 − a1 − j) 0 0

a3−ε1−1 a3−ε1−a1−j−1 (1 − ξ) (1 − η) 1F1(a1 − ε1; a3 − ε1 − a1 − j; −y(1 − ξ)(1 − η)dξdη. (2.9) 152 Ghazala Yasmin and Abdulghani Muhyi

Replacing x by −y in (2.9) and applying result (2.5) in the L.H.S. of the resultant equation we get our desired result. This completes the proof of (2.8).

3 Special Cases (i) Setting j = 0 in the results (2.1) and (2.5) yield the known results (1.9) and (1.13) respectively. (ii) Taking j = 1 in the results (2.1) and (2.5), we obtain

 1 1 1  2 a1, 2 a1 + 2 ;  2 ψ2(a1; a3, a3 + 1; −y, y) = 2F3 − y  1 1 1 a3 + 1, 2 a3 + 2 , 2 a3 + 1;

  1 a + 1 , 1 a + 1; a y 2 1 2 2 1 1  2 − 2F3 − y  (3.1) a3(a3 + 1) 1 1 3 a3 + 1, 2 a3 + 1, 2 a3 + 2 ; and     a + 1; a + 1; 1 y2 y 1 y2     φ2(a1, a1 + 1; a3; −y, y) = 1F2  + 1F2 , (3.2) 1 1 1 4 a3 1 1 1 4 2 a3, 2 a3 + 2 ; 2 a3 + 2 , 2 a3 + 1; respectively. (iii) Taking j = −1 in the results (2.1) and (2.5), we get

 1 1 1  2 a1, 2 a1 + 2 ;  2 ψ2(a1; a3, a3 − 1; −y, y) = 2F3 − y  1 1 1 a3 − 1, 2 a3, 2 a3 + 2 ;   1 a + 1 , 1 a + 1; a y 2 1 2 2 1 1  2 + 2F3 − y  (3.3) a3(a3 − 1) 1 1 1 a3, 2 a3 + 2 , 2 a3 + 1; and     a ; a ; 1 y2 y 1 y2     φ2(a1, a1 − 1; a3; −y, y) = 1F2  − 1F2 , (3.4) 1 1 1 4 a3 1 1 1 4 2 a3, 2 a3 + 2 ; 2 a3 + 2 , 2 a3 + 1; respectively. Similarly, we can derive the other special cases (for j = ±2, ±3, ±4, ±5).

Remark 3.1. Setting y = −x in (3.1)-(3.4), we get the known results ([7];17, 18,21,22).

4 Integral Formulas This section deals with some integrals involving Humbert’s functions, which are expressed in terms of Wright hypergeometric function .

Theorem 4.1. For υ, τ ∈ C and x > 0 with <(υ) > 0, <(τ) > 0, each of the following integral formulas holds true:

Z 1 x2υ−1 xτ−1 (1) xυ−1 (1 − x)2τ−1 1 − 1 − 0 3 4 SOME NEW RESULTS FOR HUMBERT’S SERIES ψ2 AND φ2 153

1 1 2 2 × ψ2 a1; a3, a3; −y 1 − x/4 (1 − x), y 1 − x/4 (1 − x) dx

  2υ 2 (a1, 2), (τ, 1); 2 Γ(υ)(Γ(a3)) 2 = 2Ψ3 − y , (4.1) 3 Γ(a1)   (a3, 1), (a3, 2), (υ + τ, 1);

Z 1 x2υ−1 xτ−1 (2) xυ−1 (1 − x)2τ−1 1 − 1 − 0 3 4

1 1 2 2 × ψ2 a1; a3, a3 + 1; −y 1 − x/4 (1 − x), y 1 − x/4 (1 − x) dx

   (a , 2), (τ, 1); 22υ (υ) (a + 1)  1 Γ Γ 3  2 = Γ(a3 + 1)2Ψ3 − y  3 Γ(a1)   (a3 + 1, 1), (a3 + 1, 2), (υ + τ, 1);   (a + 1, 2), (τ + 1 , 1); 1 2   2 − y Γ(a3) 2Ψ3 − y  , (4.2) 1  (a3 + 1, 1), (a3 + 2, 2), (υ + τ + 2 , 1);

Z 1 x2υ−1 xτ−1 (3) xυ−1 (1 − x)2τ−1 1 − 1 − 0 3 4

1 1 2 2 × ψ2 a1; a3, a3 − 1; −y 1 − x/4 (1 − x), y 1 − x/4 (1 − x) dx

   (a , 2), (τ, 1); 22υ (υ) (a )  1 Γ Γ 3  2 = Γ(a3 − 1)2Ψ3 − y  3 Γ(a1)   (a3 − 1, 1), (a3, 2), (υ + τ, 1);   (a + 1, 2), (τ + 1 , 1); 1 2   2 + y Γ(a3 − 1) 2Ψ3 − y  . (4.3) 1  (a3, 1), (a3 + 1, 2), (υ + τ + 2 , 1); Proof. By making use of the relation (1.9) in the integrand of (4.1) and then interchanging the order of the integration and summation of the resultant equation which is veriﬁed by uniform convergence of the involved series under the given conditions, gives

Z 1 x2υ−1 xτ−1 xυ−1 (1 − x)2τ−1 1 − 1 − 0 3 4

1 1 2 2 × ψ2 a1; a3, a3; −y 1 − x/4 (1 − x), y 1 − x/4 (1 − x) dx

∞ 2 k Z 1 2υ−1 (τ+k)−1 X (a1)2k (−y ) x x = xυ−1 (1−x)2(τ+k)−1 1− 1− dx. (4.4) (a3)k (a3)2k k! 3 4 k=0 0 Now, using the identity (1.22) in the above equation, we get

Z 1 x2υ−1 xτ−1 xυ−1 (1 − x)2τ−1 1 − 1 − 0 3 4 154 Ghazala Yasmin and Abdulghani Muhyi

1 1 2 2 × ψ2 a1; a3, a3; −y 1 − x/4 (1 − x), y 1 − x/4 (1 − x) dx

2υ 2 ∞ 2 k 2 Γ(υ)(Γ(a3)) X Γ(a1 + 2k) Γ(τ + k) (−y ) = , (4.5) 3 Γ(a1) Γ(a3 + k) Γ(a3 + 2k) Γ(υ + τ + k) k! k=0 which on using the deﬁnition of Wright hypergeometric function pΨq (1.5) in the above equation yields our desired result (4.1). Further, making use of the similar argument as in the above proof of the result (4.1), we can establish the integral formulas (4.2) and (4.3).

Corollary 4.2. For υ, τ ∈ C and x > 0 with <(υ) > 0, <(τ) > 0, each of the following integral formulas holds true:

Z 1 x2υ−1 xτ−1 (1) xυ−1 (1 − x)2τ−1 1 − 1 − 0 3 4

1 1 2 2 × ψ2 a1; a3, a3; −y 1 − x/4 (1 − x), y 1 − x/4 (1 − x) dx

 1 1 1  2υ τ, a1, a1 + ; 2  2 2 2 2 = B(υ, τ) 3F4 − y , (4.6) 3   1 1 1 υ + τ, a3, 2 a3, 2 a3 + 2 ; Z 1 x2υ−1 xτ−1 (2) xυ−1 (1 − x)2τ−1 1 − 1 − 0 3 4 1 1 2 2 × ψ2 a1; a3, a3 + 1; −y 1 − x/4 (1 − x), y 1 − x/4 (1 − x) dx

  1 1 1  2υ  τ, a1, a1 + ; 2   2 2 2 2 = B(υ, τ) 3F4 − y  3     1 1 1 υ + τ, a3 + 1, 2 a3 + 2 , 2 a3 + 1;   1 1 1 1 τ + 2 , a1 + 2 , a1 + 1;  a1y 1  2 2 2 − B υ, τ + 3F4 − y  , (4.7) a3(a3 + 1) 2   1 1 1 3  υ + τ + 2 , a3 + 1, 2 a3 + 1, 2 a3 + 2 ;

Z 1 x2υ−1 xτ−1 (3) xυ−1 (1 − x)2τ−1 1 − 1 − 0 3 4 1 1 2 2 × ψ2 a1; a3, a3 − 1; −y 1 − x/4 (1 − x), y 1 − x/4 (1 − x) dx

  1 1 1  2υ  τ, a1, a1 + ; 2   2 2 2 2 = B(υ, τ) 3F4 − y  3     1 1 1 υ + τ, a3 − 1, 2 a3, 2 a3 + 2 ;   1 1 1 1 τ + 2 , a1 + 2 , a1 + 1;  a1y 1  2 2 2 + B υ, τ + 3F4 − y  . (4.8) a3(a3 − 1) 2   1 1 1 1  υ + τ + 2 , a3, 2 a3 + 2 , 2 a3 + 1; Proof. By writing the r.h.s. of equation (4.6) in the series form and applying identity (1.19) to the resultant series, then after some simpliﬁcations and using identity (1.4) and deﬁnition of the generalized hypergeometric function pFq (1.1), we arrive at our assertion (4.6). Similarly we can derive integral formulas (4.7) and (4.8). SOME NEW RESULTS FOR HUMBERT’S SERIES ψ2 AND φ2 155

Remark 4.3. Using a similar argument as in the proof of Theorem 4.1, we can prove the following theorems.

Theorem 4.4. For υ, τ ∈ C and x > 0 with <(υ) > 0, <(τ) > 0, each of the following integral formulas holds true:

Z 1 x2υ−1 xτ−1 (1) xυ−1 (1 − x)2τ−1 1 − 1 − 0 3 4 1 1 × ψ2 a1; a3, a3; −yx 2 (1 − x/3), yx 2 (1 − x/3) dx

  2υ 2 (a1, 2), (υ, 1); 2 2 Γ(τ)(Γ(a3)) −4y = 2Ψ3 , (4.9) 3 Γ(a1)  9  (a3, 1), (a3, 2), (υ + τ, 1);

Z 1 x2υ−1 xτ−1 (2) xυ−1 (1 − x)2τ−1 1 − 1 − 0 3 4 1 1 × ψ2 a1; a3, a3 + 1; −yx 2 (1 − x/3), yx 2 (1 − x/3) dx

   (a , 2), (υ, 1); 22υ (τ) (a + 1)  1 −4y2 Γ Γ 3   = Γ(a3 + 1)2Ψ3  3 Γ(a1)  9  (a3 + 1, 1), (a3 + 1, 2), (υ + τ, 1);   (a + 1, 2), (υ + 1 , 1); 2y 1 2 −4y2  − Γ(a ) Ψ   , (4.10) 3 3 2 3 9  1  (a3 + 1, 1), (a3 + 2, 2), (υ + τ + 2 , 1);

Z 1 x2υ−1 xτ−1 (3) xυ−1 (1 − x)2τ−1 1 − 1 − 0 3 4 1 1 × ψ2 a1; a3, a3 − 1; −yx 2 (1 − x/3), yx 2 (1 − x/3) dx

   (a , 2), (υ, 1); 22υ (τ) (a )  1 −4y2 Γ Γ 3   = Γ(a3 − 1)2Ψ3  3 Γ(a1)  9  (a3 − 1, 1), (a3, 2), (υ + τ, 1);   (a + 1, 2), (υ + 1 , 1); 2y 1 2 −4y2  + Γ(a − 1) Ψ   . (4.11) 3 3 2 3 9  1  (a3, 1), (a3 + 1, 2), (υ + τ + 2 , 1); Theorem 4.5. For υ, τ ∈ C and x > 0 with <(υ) > 0, <(τ) > 0, each of the following integral formulas holds true:

Z 1 x2υ−1 xτ−1 (1) xυ−1 (1 − x)2τ−1 1 − 1 − 0 3 4

1 1 2 2 × φ2 a1, a1; a3; −y 1 − x/4 (1 − x), y 1 − x/4 (1 − x) dx

  2υ (a1, 1), (τ, 1); 2 Γ(υ)(Γ(a3)) 2 = 2Ψ2 y , (4.12) 3 Γ(a1)   (a3, 2), (υ + τ, 1); 156 Ghazala Yasmin and Abdulghani Muhyi

Z 1 x2υ−1 xτ−1 (2) xυ−1 (1 − x)2τ−1 1 − 1 − 0 3 4

1 1 2 2 × φ2 a1, a1 + 1; a3, −y 1 − x/4 (1 − x), y 1 − x/4 (1 − x) dx

   (a + 1, 2), (τ, 1); 22υ (υ)( (a ))  1 Γ Γ 3  2 = 2Ψ2 y  3 Γ(a1 + 1)   (a3, 2), (υ + τ, 1);   (a + 1, 1), (τ + 1 , 1); 1 2   2 + y 2Ψ2 y  , (4.13) 1  (a3 + 1, 2), (υ + τ + 2 , 1);

Z 1 x2υ−1 xτ−1 (3) xυ−1 (1 − x)2τ−1 1 − 1 − 0 3 4

1 1 2 2 × φ2 a1, a1 − 1; a3, y 1 − x/4 (1 − x), y 1 − x/4 (1 − x) dx

   (a , 2), (τ, 1); 22υ (υ)( (a ))  1 Γ Γ 3  2 = 2Ψ2 y  3 Γ(a1)   (a3, 2), (υ + τ, 1);   (a , 1), (τ + 1 , 1); 1 2   2 − y 2Ψ2 y  . (4.14) 1  (a3 + 1, 2), (υ + τ + 2 , 1);

Theorem 4.6. For υ, τ ∈ C and x > 0 with <(υ) > 0, <(τ) > 0, each of the following integral formulas holds true:

Z 1 x2υ−1 xτ−1 (1) xυ−1 (1 − x)2τ−1 1 − 1 − 0 3 4

1 1 × φ2 a1, a1; a3; −yx 2 (1 − x/3), yx 2 (1 − x/3) dx

  2υ (a1, 1), (υ, 1); 2 2 Γ(τ)(Γ(a3) 4y = 2Ψ2 , (4.15) 3 Γ(a1)  9  (a3, 2), (υ + τ, 1);

Z 1 x2υ−1 xτ−1 (2) xυ−1 (1 − x)2τ−1 1 − 1 − 0 3 4 1 1 × φ2 a1, a1 + 1; a3, −yx 2 (1 − x/3), yx 2 (1 − x/3) dx

   (a + 1, 1), (υ, 1); 22υ (τ) (a )  1 4y2 Γ Γ 3   = 2Ψ2  3 Γ(a1 + 1)  9  (a3, 2), (υ + τ, 1);   (a + 1, 1), (υ + 1 , 1); 2y 1 2 4y2  + Ψ   , (4.16) 3 2 2 9  1  (a3 + 1, 2), (υ + τ + 2 , 1); SOME NEW RESULTS FOR HUMBERT’S SERIES ψ2 AND φ2 157

Z 1 x2υ−1 xτ−1 (3) xυ−1 (1 − x)2τ−1 1 − 1 − 0 3 4

1 1 × φ2 a1, a3 − 1; a3, −yx 2 (1 − x/3), yx 2 (1 − x/3) dx

   (a , 1), (υ, 1); 22υ (τ) (a )  1 4y2 Γ Γ 3   = 2Ψ2  3 Γ(a1)  9  (a3, 2), (υ + τ, 1);   (a , 1), (υ + 1 , 1); 2y 1 2 4y2  − Ψ   . (4.17) 3 2 2 9  1  (a3 + 1, 2), (υ + τ + 2 , 1); We conclude this paper by remarking that this approach is general and can be applied to some other multivariate hypergeometric functions and is a problem for further research.

References

[1] W. N. Bailey, Generalized Hypergeometric Series, Cambridge: Cambridge Univ. Press, (1935), Reprinted by Stechert Hafner, New York, (1964). [2] Yu. A. Brychkov, Handbook of Special Functions, Derivatives, Integrals, Series and Other Formulas, CRC Press, Taylar and Francis Group, A Chapman and Hall Book, (2008). [3] J. L. Burchnall and T. W. Chaundy, Expansions of Appell’s double hypergeometric functions, Quart. J. Math. (Oxford Ser.) 11, 249–270 (1940).

[4] J. Choi, Contiguous Extensions of Dixon’s Theorem on the Sum of a 3F2, J. Inequal. Appl. 2010, Article ID 589618 (2010). [5] J. Choi and A. Hasanov, Applications of the operator H(α, β) to the Humbert double hypergeometric functions, Comput. Math. Appl. 61, 663–671 (2011). [6] J. Choi and A. K. Rathie, Certain new generating relations for products of two Laguerre polynomials, Commun. Korean Math. Soc. 30 (3), 191–200 (2015).

[7] J. Choi and A. K. Rathie, Certain summation formulas for Humbert’s double hypergeometric series ψ2 and φ2, Korean Math. Soc. 30 (4), 439–445 (2015). [8] J. Choi and A. K. Rathie, Generalizations of two summation formulas for the generalized hypergeometric function of higher order due to Exton, Commun. Korean Math. Soc. 25 (3), 385–389 (2010). [9] J. Choi and A. K. Rathie, Reducibility of Certain Kampé De Fériet Function with an Application to Generating relations for products of two Laguerre polynomials, Filomat 30:7, 2059–2066 (2016).

[10] J. L. Lavoie, F. Grondin and A. K. Rathie, Generalizations of Whipple’s theorem on the sum of a 3F2, J. Comput. Appl. Math. 72 (2), 293–300 (1996). [11] J. L. Lavoie and G. Trottier, On the sum of certain Appell’s series, Ganita 20 (1), 31–32 (1969). [12] G. Lohofer, Theory of an electromagnetically deviated metal sphere.1:absorbed power, SIAM J. Appl. Math. 49, 567–581 (1989).

[13] V. V. Manako, A connection formula between double hypergeometric series ψ2 and φ3, Int. Trans. Spec. Fun. 23 (7), 503–508 (2012). [14] A. M. Mathai and R. K. Saxena, Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences, Springer-Verlag, Berlin, Heidelberg, New York, (1973). N [15] A. W. Niukkanen, Generalised hypergeometric series F (x1, ..., xN ) arising in physical and quantum chemical applications, J. Phys. A: Math. Gen. 16, 1813–1825 (1983). [16] E. D. Rainville, Special Functions, Macmillan Company, New York, (1960).

[17] A. K. Rathie, On representation of Humbert’s double hypergeometric series φ2 in a series of Gauss’s 2F1 function, arXiv: 1312.0064v1 (2016). [18] N. Shekhawat, J. Choi, A. K. Rathie and Om Prakash, On a new class of series identities, Honam Math. J. 37(3), 339–352 (2015). 158 Ghazala Yasmin and Abdulghani Muhyi

[19] H. M. Srivastava and B. R. K. Kashyap, Special Functions in Queuing Theory and Related Stochastic Processes, Academic Prees, New York, London, San Francisco, (1982). [20] H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, (1984). [21] Wei et al., Certain transformations for multiple hypergeometric functions, Advances in Difference Equa- tions, 2013:360 (2013).

Author information Ghazala Yasmin and Abdulghani Muhyi, Department of Applied Mathematics, Faculty of Engineering and Technology, Aligarh Muslim University, Aligarh-202002, India. E-mail: [email protected], [email protected]

Received: April 10, 2018. Accepted: September 9, 2018.