# Some Formulas for New Quadruple Hypergeometric Functions

Authors: Jihad A. Younis; Hassen Aydi; Ashish Verma
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we aim to introduce six new quadruple hypergeometric functions. Then, we investigate certain formulas and representations for these functions such as symbolic formulas, differential formulas, and integral representations.


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## Body

## 1. Introduction

Hypergeometric functions of several variables play an important role in diverse areas of science and engineering. The developments in applied mathematics, mathematical physics, chemistry, combinatorics, statistics, numerical analysis, and other areas have led to increasing interest in the study of multiple hypergeometric functions. Many authors have studied a number of formulas involving hypergeometric functions (see, e.g., [1–6]).In [7], Exton presented twenty-one complete hypergeometric functions in four variables denoted by symbols K1,K2,…,K21. In [8], Sharma and Parihar defined eighty-three complete quadruple hypergeometric functions, namely, F14,F24,…,F834. Bin-Saad and Younis [9] gave thirty new quadruple hypergeometric functions given by X14,X24,…,X304. In [10], the authors discovered the existence of twenty additional complete hypergeometric functions in four variables X314,X324,…,X504. Each quadruple hypergeometric function in [7–10] is of the form(1)X4.=∑m,n,p,q=0∞Ωm,n,p,qxmm!ynn!zpp!uqq!,where Ωm,n,p,q is a certain sequence of complex parameters, and there are twelve parameters in each series of X4. (eight a′s and four c′s ). The 1st, 2nd, 3rd, and 4th parameters in X4. are connected with integers m,n,p, and q, respectively. Each repeated parameter in the series X4. points out a term with double parameters in Ωm,n,p,q. For example, X4a1,a1,a2,a2,a3,a3,a4,a5 means that a1m+na2p+qa3m+na4pa5q includes the term. Similarly, X4a1,a1,a1,a2,a1,a1,a2,a3 points out the term a12m+2n+pa2p+qa3q, and X4a1,a1,a2,a3,a1,a2,a3,a4 shows the existence of the term a12m+na2n+pa3p+qa4q. Thus, it is possible to form various combinations of indices. There seems to be no way of independently establishing the number of distinct Gaussian hypergeometric series for any given integer n≥2 without explicitly stating all such series. Thus, in every situation with n=4, one ought to begin by actually constructing the set just as in the case n=3 (see [11]).By using the conventions and notations above, we now introduce further quadruple hypergeometric functions as follows:(2)X854a1,a1,a2,a3,a1,a2,a3,a4;c1,c2,c3,c1;x,y,z,u=∑m,n,p,q=0∞a12m+na2n+pa3p+qa4qc1m+qc2nc3pxmm!ynn!zpp!uqq!,X864a1,a1,a2,a3,a1,a2,a3,a4;c2,c1,c3,c1;x,y,z,u=∑m,n,p,q=0∞a12m+na2n+pa3p+qa4qc1n+qc2mc3pxmm!ynn!zpp!uqq!,X874a1,a1,a2,a3,a1,a2,a3,a4;c1,c2,c1,c2;x,y,z,u==∑m,n,p,q=0∞a12m+na2n+pa3p+qa4qc1m+pc2n+qxmm!ynn!zpp!uqq!,X884a1,a1,a2,a3,a1,a2,a3,a4;c1,c2,c2,c1;x,y,z,u=∑m,n,p,q=0∞a12m+na2n+pa3p+qa4qc1m+qc2n+pxmm!ynn!zpp!uqq!,X894a1,a1,a2,a3,a1,a2,a3,a4;c1,c2,c1,c1;x,y,z,u=∑m,n,p,q=0∞a12m+na2n+pa3p+qa4qc1m+p+qc2nxmm!ynn!zpp!uqq!,X904a1,a1,a2,a3,a1,a2,a3,a4;c,c,c,c;x,y,z,u=∑m,n,p,q=0∞a12m+na2n+pa3p+qa4qcm+n+p+qxmm!ynn!zpp!uqq!,for(3)x<14,y<1,z<1,u<1.Here,am is the Pochhammer symbol defined (for a,m∈ℂ), in terms of the familiar Gamma function Γ, by (see, e.g., [11], p. 2 and p. 5)(4)am≔Γa+mΓa,a+m∈ℂ\ℤ0−=1,m=0,aa+1…a+m−1,m=n∈ℕ,where ℂ,ℤ0−, and ℕ denote the sets of complex numbers, nonpositive integers, and positive integers, respectively.We recall the Gauss hypergeometric function [12] which is defined by(5)F21a,b;c;x=∑n=0∞anbncnxnn!,x<1.Appell’s double hypergeometric functionF2 is defined as follows [13]:(6)F2a,b,c;d,e;x,y=∑m,n=0∞am+nbmcndmenxmm!ynn!.In [14], Exton established twenty distinct triple hypergeometric functions, which are denoted by X1,X2,…,X20. We introduce the definitions of five of these functions in the following:(7)X15a1,a2,a3;c2,c1;x,y,z=∑m,n,p=0∞a12m+na2n+pa3pc1n+pc2mxmm!ynn!zpp!,X16a1,a2,a3;c1,c2;x,y,z=∑m,n,p=0∞a12m+na2n+pa3pc1m+pc2nxmm!ynn!zpp!,X17a1,a2,a3;c1,c2,c3;x,y,z=∑m,n,p=0∞a12m+na2n+pa3pc1mc2nc3pxmm!ynn!zpp!,X18a1,a2,a3,a4;c;x,y,z=∑m,n,p=0∞a12m+na2na3pa4pcm+n+pxmm!ynn!zpp!,X20a1,a2,a3,a4;c1,c2;x,y,z=∑m,n,p=0∞a12m+na2na3pa4pc1m+pc2nxmm!ynn!zpp!.The Lauricella functions of three variablesFM,FN,Fp,FS, and FT are defined in [11, 15]:(8)FMa1,a2,a2,b1,b2,b1;c1,c2,c2;x,y,z=∑m,n,p=0∞a1ma2n+pb1m+pb2nc1mc2n+pxmm!ynn!zpp,FNa1,a2,a3,b1,b2,b1;c1,c2,c2;x,y,z=∑m,n,p=0∞a1ma2na3pb1m+pb2nc1mc2n+pxmm!ynn!zpp,FPa1,a2,a1,b1,b1,b2;c1,c2,c2;x,y,z=∑m,n,p=0∞a1m+pa2nb1m+nb2pc1mc2n+pxmm!ynn!zpp,FSa1,a2,a2,b1,b2,b3;c,c,c;x,y,z=∑m,n,p=0∞a1ma2n+pb1mb2nb3pcm+n+pxmm!ynn!zpp,FTa1,a2,a2,b1,b2,b1;c,c,c;x,y,z=∑m,n,p=0∞a1ma2n+pb1m+pb2ncm+n+pxmm!ynn!zpp.The structure of this paper is as follows. In Sections2 to 5, we obtain several symbolic formulas, differentiation formulas, operator formulas, and integral representations for the hypergeometric functions of four variables X854,X864,…,X904.


## 2. Symbolic Formulas

First of all, we recall the following symbolic operators (see [16]):(9)Dδmδs=Γs+1Γs−m+1δs−m,(10)Dδ−mδs=Γs+1Γs+m+1δs+m,for(11)m∈ℕ∪0,s∈ℂ−−1,−2,…,where Dδ and Dδ−1 are the derivative and integral operator, respectively.Now, we find the following formulas.Theorem 1.
The following results hold true:(12)1−Dα1Dα2β−1Dβ−1γ−1Dγ−1α1α2u−aX17a1,a2,a3;c1,c2;βx,y,α1z×α1a3−1α2a4−1βc1−1γa−1=α1a3−1α2a4−1βc1−1γa−1X854a1,a1,a2,a3,a1,a2,a3,a4c1,c2,c3,c1;βx,y,α1z,u;,(13)1−Dα1Dα2β−1Dβ−1γ−1Dγ−1α1α2z−aX20a1,a2,a3,a4;c1,c2;x,α1y,α2u×α1a2−1α2a3−1βc3−1γa−1=α1a2−1α2a3−1βc3−1γa−1X854a1,a1,a2,a3,a1,a2,a3,a4;c1,c2,c3,c1;x,α1y,z,α2u.Proof.
To prove the result in equality (12) asserted in Theorem 1, let ∅ denote the left-hand side of equality (12). Then, employing the series representation of x17 and by using (9) and (10), we have(14)∅=∑m,n,p,q=0∞a12m+na2n+pa3paqβ−qγ−qc1mc2nc3pm!n!p!q!xmynzpuq×Dα1qDα2qDβ−qDγ−qα1a3+p+q−1α2a4+q−1βc+m+q−1γa−1=α1a3−1α2a4−1βc1−1γa−1∑m,n,p,q=0∞a12m+na2n+pa3p+qa4qc1m+qc2nc3pβxmm!ynn!α1zpp!uqq!=α1a3−1α2a4−1βc1−1γa−1X854a1,a1,a2,a3,a1,a2,a3,a4;c1,c2,c3,c1;x,y,z,u,which completes the proof. Similarly, one can prove formulas (13) and (20).Theorem 2.
The following results hold true:(15)1−Dα2β−1Dβ−1γ−1Dγ−1α2x−aFPa3,a3,a1,a2,a2,a4;c3,c1,c1;z,αy,u×αa1−1βc2−1γa−1=αa1−1βc2−1γa−1X864a1,a1,a2,a3,a1,a2,a3,a4;c2,c1,c3,c1;x,αy,z,u,1−Dα1Dα2β−1Dβ−1γ−1Dγ−1α1α2y2−αF1a12,a1+12;c2;4α12x×F2a3,a2,a4;c3,c1;α2z,βuα1a1−1α2a2−1βc1−1γa−1=α1a1−1α2a2−1βc1−1γa−1X864a1,a1,a2,a3,a1,a2,a3,a4;c2,c1,c3,c1;α12x,y,α2z,βu.Theorem 3.
The following results hold true:(16)1−Dα1Dα2β−1Dβ−1γ−1Dγ−1α1α2u−aX16a1,a2,a3;c1,c2;x,βy,α1z×α1a3−1α2a4−1βc2−1γa−1=α1a3−1α2a4−1βc2−1γa−1X874a1,a1,a2,a3,a1,a2,a3,a4;c1,c2,c1,c2;x,βy,α1z,u,1−Dα1Dα2β−1Dβ−1γ−1Dγ−1α1α2y−aFNa4,a12,a2,a3,a3,a1+12;c2,c1,c1;βu,4α12x,α2z×α1a1−1α2a2−1βc2−1γa−1=α1a1−1α2a2−1βc2−1γa−1X874a1,a1,a2,a3,a1,a2,a3,a4;c1,c2,c1,c2;α12x,y,α2z,βu.Theorem 4.
The following results hold true:(17)1−Dα1Dα2β−1Dβ−1γ−1Dγ−1α1α2u−aX15a1,a2,a3;c1,c2;βx,y,α1z×α1a3−1α2a4−1βc1−1γa−1=α1a3−1α2a4−1βc1−1γa−1X884a1,a1,a2,a3,a1,a2,a3,a4;c1,c2,c2,c1;βx,y,α1z,u,1−Dα2β−1Dβ−1γ−1Dγ−1α2x−aFMa4,a2,a2,a3,a3,a1;c1,c2,c2;βu,αy,z×αa1−1βc1−1γa−1=αa1−1βc1−1γa−1X884a1,a1,a2,a3,a1,a2,a3,a4;c1,c2,c2,c1;x,αy,z,βu.Theorem 5.
The following results hold true:(18)1−Dα1Dα2β−1Dβ−1γ−1Dγ−1α1α2u−aX16a1,a2,a3;c1,c2;βx,y,α1βz×α1a3−1α2a4−1βc1−1γa−1=α1a3−1α2a4−1βc1−1γa−1X894a1,a1,a2,a3,a1,a2,a3,a4;c1,c2,c1,c1;βx,y,α1βz,u,1−Dα1Dα2β−1Dβ−1γ−1Dγ−1α1α2y−aFSa12,a3,a3,a1+12,a2,a4;c1,c1,c1;4α1x,α2z,u×α1a1−1α2a2−1βc2−1γa−1=α1a1−1α2a2−1βc2−1γa−1X894a1,a1,a2,a3,a1,a2,a3,a4;c1,c2,c1,c1;α1x,y,α2z,u.Theorem 6.
The following results hold true:(19)1−Dα1Dα2β−1Dβ−1γ−1Dγ−1α1α2z−aX18a1,a2,a3,a4;c;βx,α1βy,α2βu×α1a2−1α2a3−1βc−1γa−1=α1a2−1α2a3−1βc−1γa−1X904a1,a1,a2,a3,a1,a2,a3,a4;c,c,c,c;βx,α1βy,z,α2βu,(20)1−Dα2β−1Dβ−1γ−1Dγ−1α2x−aFTa4,a2,a2,a3,a1,a3;c,c,c;βu,αβy,βz×αa1−1βc−1γa−1=αa1−1βc−1γa−1X904a1,a1,a2,a3,a1,a2,a3,a4c,c,c,c;x,αβy,βz,βu;.


## 3. Differentiation Formulas

The results of this section can be derived from formula (9) by a direct evaluation.Theorem 7.
The following derivative formulas hold true:(21)Dw1a1−cDw2a2−c′w1a1−1w2a2−1X854c,c,c′,a3,c,c′,a3,a4;c1,c2,c3,c1;w12x,w1w2y,w2z,u=Γa1Γa2ΓcΓc′w1c−1w2c′−1X854a1,a1,a2,a3,a1,a2,a3,a4;c1,c2,c3,c1;w12x,w1w2y,w2z,u,Dw1a2−cDw2a3−c′w1a2−1w2a3−1X854a1,a1,c,c′,a1,c,c′,a4;c1,c2,c3,c1;x,w1y,w1w2z,w2u=Γa2Γa3ΓcΓc′w1c−1w2c′−1X854a1,a1,a2,a3,a1,a2,a3,a4;c1,c2,c3,c1;x,w1y,w1w2z,w2u,Dw1a3−cDw2a4−c′w1a3−1w2a4−1X854a1,a1,a2,c,a1,a2,c,c′;c1,c2,c3,c1;x,y,w1z,w1w2u=Γa3Γa4ΓcΓc′w1c−1w2c′−1X854a1,a1,a2,a3,a1,a2,a3,a4;c1,c2,c3,c1;x,y,w1z,w1w2u.Theorem 8.
The following differentiation formulas hold:(22)Dxa1−cxa1−1X864c,c,a2,a3,c,a2,a3,a4;c2,c1,c3,c1;x2,xy,z,u=Γa1Γcxc−1X864a1,a1,a2,a3,a1,a2,a3,a4;c2,c1,c3,c1;x2,xy,z,u,Dxa1−cDya2−c′xa1−1ya2−1X864c,c,c′,a3,c,c′,a3,a4;c2,c1,c3,c1;x2,xy,yz,u=Γa1Γa2ΓcΓc′xc−1yc′−1X864a1,a1,a2,a3,a1,a2,a3,a4;c2,c1,c3,c1;x2,xy,yz,u,Dua4−cua4−1X864a1,a1,a2,a3,a1,a2,a3,c;c2,c1,c3,c1;x,y,z,u=Γa4Γcuc−1X864a1,a1,a2,a3,a1,a2,a3,a4;c2,c1,c3,c1;x,y,z,u.Theorem 9.
The following derivative formulas hold true:(23)Dwa2−cDza3−c′wa2−1za3−1X874a1,a1,c,c′,a1,c,c′,a4;c1,c2,c1,c2;x,wy,wz,uz=Γa2Γa3ΓcΓc′wc−1zc′−1X874a1,a1,a2,a3,a1,a2,a3,a4;c1,c2,c1,c2;x,wy,wz,uz,Dwa1−cDya2−c′Dza3−c″wa1−1ya2−1za3−1X874c,c,c′,c″,c,c′,c″,a4;c1,c2,c1,c2;w2x,wy,yz,uz=Γa1Γa2Γa3ΓcΓc′Γc″wc−1yc′−1zc″−1X874a1,a1,a2,a3,a1,a2,a3,a4;c1,c2,c1,c2;w2x,wy,yz,uz,Dw1a1−cDw2a4−c′w1a1−1w2a4−1X874c,c,a2,a3,c,a2,a3c′;c1,c2,c1,c2;w12x,w1y,z,w2u=Γa1Γa4ΓcΓc′w1c−1w2c′−1X874a1,a1,a2,a3,a1,a2,a3,a4;c1,c2,c1,c2;w12x,w1y,z,w2u.Theorem 10.
The following derivative formulas hold true:(24)Dya2−cya2−1X884a1,a1,c,a3,a1,c,a3,a4;c1,c2,c2,c1;x,y,yz,u=Γa2Γcyc−1X884a1,a1,a2,a3,a1,a2,a3,a4;c1,c2,c2,c1;x,y,yz,u,Dza2−cza2−1X884a1,a1,c,a3,a1,c,a3,a4;c1,c2,c2,c1;x,yz,z,u=Γa2Γczc−1X884a1,a1,a2,a3,a1,a2,a3,a4;c1,c2,c2,c1;x,yz,z,u,Dw1a1−cDw2a2−c′Dw3a3−c′′Dw4a4−c′′′w1a1−1w2a2−1w3a3−1w4a4−1X884c,c,c′,c″,c,c′,c″,c‴;c1,c2,c2,c1;w12x,w1w2y,w2w3z,w3w4u=Γa1Γa2Γa3Γa4ΓcΓc′Γc″Γc‴w1c−1w2c′−1w3c″−1w4c‴−1×X884a1,a1,a2,a3,a1,a2,a3,a4;c1,c2,c2,c1;w12x,w1w2y,w2w3z,w3w4u.Theorem 11.
The following derivative formulas hold true:(25)Dya1−cya1−1X894c,c,a2,a3,c,a2,a3,a4;c1,c2,c1,c1;xy2,y,z,u=Γa1Γcyc−1X884a1,a1,a2,a3,a1,a2,a3,a4;c1,c2,c1,c1;xy2,y,z,u,Dya1−cDwa2−c′ya1−1wa2−1X894c,c,c′,a3,c,c′,a3,a4;c1,c2,c1,c1;xy2,wy,wz,u=Γa1Γa2ΓcΓc′yc−1wc′−1X894a1,a1,a2,a3,a1,a2,a3,a4;c1,c2,c1,c1;xy2,wy,wz,u,Dxa1−cDw1a2−c″Dza3−c‴Dw2a4−c‴xa1−1w1a2−1za3−1w2a4−1X894c,c,c′,c″,c,c′,c″,c″;c1,c2,c1,c1;x2,w1xy,w1z,w2uz=Γa1Γa2Γa3Γa4ΓcΓc′Γc″Γc‴xc−1w1c′−1zc″−1w2c‴−1×X894a1,a1,a2,a3,a1,a2,a3,a4;c1,c2,c1,c1;x2,w1xy,w1z,w2uz.Theorem 12.
The following derivative formulas hold true:(26)Dza2−c′Dua3−c′′za2−1ua3−1X904a1,a1,c′,c″,a1,c′,c″,a4;c,c,c,c;x,yz,uz,u=Γa2Γa3ΓcΓc′zc′−1uc″−1X904a1,a1,a2,a3,a1,a2,a3,a4;c,c,c,c;x,yz,uz,u,Dya2−c′Dwa3−c″Dua4−c‴ya2−1wa3−1ua4−1X904a1,a1,c′,c″,a1,c′,c″,c‴;c,c,c,c;x,y,wyz,wu=Γa2Γa3Γa4Γc′Γc″Γc‴yc′−1wc′′−1uc‴−1X904a1,a1,a2,a3,a1,a2,a3,a4;c,c,c,c;x,y,wyz,wu,Dw1a1−c′Dza3−c′′Dw2a4−c′′′w1a1−1za3−1w2a4−1X904c′,c′,a2,c″,c′,a2,c″,c‴;c,c,c,c;w12x,w1y,z,w2uz=Γa1Γa3Γa4Γc′Γc″Γc‴w2c−1zc″−1w2c‴−1X904a1,a1,a2,a3,a1,a2,a3,a4;c,c,c,c;w12x,w1y,z,w2uz.


## 4. Integral Representations

In this section, we give integral representations of Laplace type for our new hypergeometric functions of four variables.Theorem 13.
Each of the following integral representations holds true:(27)X854a1,a1,a2,a3,a1,a2,a3,a4;c1,c2,c3,c1;x,y,z,u=1Γa1Γa3∫0∞∫0∞×e−s+tsa1−1ta3−1Φ3a4;c1;tu,s2xΨ2a4;c2,c3;sy,tzdsdtRea1>0,Rea3>0,(28)X864a1,a1,a2,a3,a1,a2,a3,a4;c2,c1,c3,c1;x,y,z,u=1Γa1Γa2Γa3∫0∞∫0∞∫0∞×e−s+t+vsa1−1ta2−1va3−1Φ3a4;c1;vu,styF01−;c2;s2x0F1−;c3;tvzdsdtdvRea1>0,Rea2>0,Rea3>0,(29)X874a1,a1,a2,a3,a1,a2,a3,a4;c1,c2,c1,c2;x,y,z,u=1Γa1Γa2Γa3∫0∞∫0∞∫0∞×e−s+t+vsa1−1ta2−1va3−1F01−;c1;s2x+tvzΦ3a4;c2;vu,stydsdtdvRea1>0,Rea2>0,Rea3>0,(30)X884a1,a1,a2,a3,a1,a2,a3,a4;c1,c2,c2,c1;x,y,z,u=1Γa1Γa3∫0∞∫0∞×e−s+tsa1−1ta3−1Φ3a4;c1;tu,s2xF11a2;c2;sy+tzdsdtRea1>0,Rea3>0,(31)X894a1,a1,a2,a3,a1,a2,a3,a4;c1,c2,c1,c1;x,y,z,u=1Γa1Γa2Γa4∫0∞∫0∞∫0∞×e−s+t+vsa1−1ta2−1va4−1Φ3a3;c1;tz+vu,s2xF01−;c2;stydsdtdvRea1>0,Rea2>0,Rea4>0,(32)X904a1,a1,a2,a3,a1,a2,a3,a4;c,c,c,c;x,y,z,u=1Γa1Γa3∫0∞∫0∞×e−s+tsa1−1ta3−1Φ33a2,a4;c;sy+tz,tu,s2xdsdtRea1>0,Rea3>0,where F01,F11,Ψ2,Φ3, and Φ33 are the confluent hypergeometric functions defined by (see [11])(33)F01−;c;x=∑m=0∞1cmxmm!,F11a;c;x=∑m=0∞amcmxmm!,Ψ2a;b,c;x,y=∑m=0∞am+nbmcnxmm!ynn!,Φ3a;c;x,y=∑m,n=0∞amcm+nxmm!ynn!,Φ33a,b;c;x,y,z=∑m,n=0∞ambncm+n+pxmm!ynn!zpp!.Proof.
It is noted that each of the integral representations (27) to (32) can be proved mainly by expressing the series definition of the involved special functions in each integrand, changing the order of the integral sign and the summation, and finally using the following well-known integral formula [12, 17]:(34)Γz=∫0∞e−ttz−1dt,ℜz>0.


## 5. Operator Formulas

Here, we establish some operator identities for functionsX854,X864,…,X904. We begin by recalling the following reciprocally inverse operators (see [3, 18]):(35)Ht1,…,tia,b≔ΓbΓa+δ1+⋯+δiΓaΓb+δ1+⋯+δi=∑k1,…,ki=0∞b−ak1+⋯+ki−δ1k1…−δikibk1+⋯+kik1!…ki!,H¯t1,…,tia,b≔ΓaΓa+δ1+⋯+δiΓbΓa+δ1+⋯+δi=∑k1,…,ki=0∞b−ak1+⋯+ki−δ1k1…−δiki1−a−δ1−⋯−δik1+⋯+kik1!…ki!,where δj≔tj∂/∂tj,j=1,…,i;i∈ℕ≔1,2,3,….Theorem 14.
The following identities hold true:(36)X854a1,a1,a2,a3,a1,a2,a3,a4;c1,c2,c3,c1;x,y,z,u=Hy,za2,aX854a1,a1,a,a3,a1,a,a3,a4;c1,c2,c3,c1;x,y,z,u,(37)X854a1,a1,a2,a3,a1,a2,a3,a4;c1,c2,c3,c1;x,y,z,u=Hzc,c3X854a1,a1,a2,a3,a1,a2,a3,a4;c1,c2,c,c1;x,y,z,u.Theorem 15.
The following identities hold true:(38)X864a1,a1,a2,a3,a1,a2,a3,a4;c2,c1,c3,c1;x,y,z,u=H¯z,ua,a3X864a1,a1,a2,a,a1,a2,a,a4;c2,c1,c3,c1;x,y,z,u,(39)X864a1,a1,a2,a3,a1,a2,a3,a4;c2,c1,c3,c1;x,y,z,u=H¯xc2,cX864a1,a1,a2,a3,a1,a2,a3,a4;c,c1,c3,c1;x,y,z,u.Theorem 16.
The following identities hold true:(40)X874a1,a1,a2,a3,a1,a2,a3,a4;c1,c2,c1,c2;x,y,z,u=Hua4,aX874a1,a1,a2,a3,a1,a2,a3,a;c1,c2,c1,c2;x,y,z,u,(41)X874a1,a1,a2,a3,a1,a2,a3,a4;c1,c2,c1,c2;x,y,z,u=Hx,zc,c1Hy,uc′,c2X874a1,a1,a2,a3,a1,a2,a3,a4;c,c′,c,c′;x,y,z,u.Theorem 17.
The following identities hold true:(42)X884a1,a1,a2,a3,a1,a2,a3,a4;c1,c2,c2,c1;x,y,z,u=H¯y,za,a2H¯ua′,a4X884a1,a1,a,a3,a1,a,a3,a′;c1,c2,c2,c1;x,y,z,u,(43)X884a1,a1,a2,a3,a1,a2,a3,a4;c1,c2,c2,c1;x,y,z,u=Hx,uc,c1X884a1,a1,a2,a3,a1,a2,a3,a4;c,c2,c2,c;x,y,z,u.Theorem 18.
The following identities hold true:(44)X894a1,a1,a2,a3,a1,a2,a3,a4;c1,c2,c1,c1;x,y,z,u=Hx,z,uc,c1X894a1,a1,a2,a3,a1,a2,a3,a4;c,c2,c,c;x,y,z,u,(45)X894a1,a1,a2,a3,a1,a2,a3,a4;c1,c2,c1,c1;x,y,z,u=H¯x,z,uc1,cX894a1,a1,a2,a3,a1,a2,a3,a4;c,c2,c,c;x,y,z,u.Theorem 19.
The following identities hold true:(46)X904a1,a1,a2,a3,a1,a2,a3,a4;c,c,c,c;x,y,z,u=H¯ua,a4X904a1,a1,a2,a3,a1,a2,a3,a;c,c,c,c;x,y,z,u,(47)X904a1,a1,a2,a3,a1,a2,a3,a4;c,c,c,c;x,y,z,u=Hz,ua3,aX904a1,a1,a2,a,a1,a2,a,a4;c,c,c,c;x,y,z,u.Proof.
Relations (37) to (47) can be proved by means of Mellin and Mellin–Barnes integral representation methods for hypergeometric functions (see [19]). The details of proofs are omitted.



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