# Linearly Independent Solutions and Integral Representations for Certain Quadruple Hypergeometric Function

Authors: Anvar Hasanov; Jihad Younis; Hassen Aydi
Journal: Journal of Function Spaces
 (2021)

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## Abstract

Recently, hypergeometric functions of four variables are investigated by Bin-Saad and Younis. In this manuscript, our goal is to initiate a new quadruple hypergeometric function denoted byX844, and then, we ensure the existence of solutions of systems of partial differential equations for this function. We also establish some integral representations involving the quadruple hypergeometric function X844.


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## Body

## 1. Introduction

Special functions, in recent years, are a piece of research that turned out to be very attractive to many scholars, hunting generalisations which are almost always evoked by applications. Hypergeometric functions of several variables have many applications in mathematical physics, statistical sciences, physics, dynamics, quantum mechanics, chemistry, and engineering (see, e.g., [1–7]). Multivariable hypergeometric functions occur in diverse areas of mathematics such as approximation theory, partition theory, representation theory, group theory, mirror symmetry, and algebraic geometry. They possess important properties such as recurrence and explicit relations, summation formulas, symmetric and convolution identities, and algebraic properties. Furthermore, multidimensional hypergeometric functions are used to solve boundary value problems (Dirichlet problem, Neumann problem, and Holmgren problem) for multidimensional degenerate differential equations (see [8–12]).In [13], Bin-Saad and Younis introduced several integral representations of Euler type and Laplace type for new hypergeometric functions in four variables. The authors, in [14], defined four new quadruple hypergeometric functions, namely, X804,X814,X824,andX834, and they obtained fractional derivative formulas, integral representations, and operational formulas for these quadruple hypergeometric functions. More recently, Younis et al. [15] introduced and studied further quadruple hypergeometric functions denoted by X854,X864,⋯,X904. Each quadruple function in [13–15] can be expressed as
(1)X4·=∑m,n,p,q=0∞Θm,n,p,qxmm!ynn!zpp!uqq!,where Θm,n,p,q is a sequence of complex parameters and there are twelve parameters in every series X4· (eight a’s and four c’s). The 1st, 2nd, 3rd, and 4th parameters in X4· are connected with the integers m,n,p, and q, respectively. Every repeated parameter in the series X4· points out a term with double parameters in Θm,n,p,q. Hence, it is possible to form various combinations of indices. It seems that there is no way to establish independently the number of distinct Gaussian hypergeometric series for each arbitrary integer n≥2 without giving explicitly all such series. Hence, in each situation with n=4, one ought to start with actually building the set like the case n=3 (refer to [16]).Motivated by the works [13–15], we define here the following quadruple hypergeometric function:
(2)X844ℓ1,ℓ2,ℓ3,ℓ4;j1,j2,j3;x,y,z,t=∑m,n,p,q=0∞ℓ12m+nℓ2n+pℓ3p+qℓ4qj1m+pj2nj3qxmm!ynn!zpp!tqq!x<14,y<1<z<1<t<1,where ℓn is the well-known Pochhammer symbol given as
(3)ℓn≔1,n=0,ℓℓ+1⋯ℓ+n−1,n∈ℕ≔1,2,⋯.Throughout this paper,ℕ, ℤ−, and ℂ denote the sets of positive integers, negative integers, and complex numbers, respectively. Also,
(4)ℕ0≔ℕ∪0,ℤ0−≔ℤ−∪0.Recently, various interesting hypergeometric functions in several variables have been investigated by many authors (see, e.g., [17–24]). In Section 2, we show how to find the linearly independent solutions of partial differential equations satisfied by the function X844. Section 3 is aimed at presenting some integral representations of Euler type for our quadruple function.


## 2. Solving Systems of Partial Differential Equations

Following the theory of multiple hypergeometric functions [25], the system of partial differential equations for the quadruple hypergeometric function X844 is given as follows:
(5)j1+x∂∂x+z∂∂zx∂∂x+1x−1u−ℓ1+2x∂∂x+2y∂∂y+1ℓ1+2x∂∂x+2y∂∂yu=0,j2+y∂∂yy∂∂y+1y−1u−ℓ1+2x∂∂x+2y∂∂yℓ2+2x∂∂x+y∂∂yu=0,j1+x∂∂x+z∂∂zz∂∂z+1z−1u−ℓ2+y∂∂y+z∂∂zℓ3+z∂∂z+t∂∂tu=0,j3+t∂∂tt∂∂t+1t−1u−ℓ3+z∂∂z+t∂∂tℓ4+t∂∂tu=0,where u=X844ℓ1,ℓ2,ℓ3,ℓ4;j1,j2,j3;x,y,z,t.Starting from (5) and by making use of some elementary calculations, we define the system of second-order partial differential equations:
(6)x1−4xuxx−4xyuxy+zuxz−y2uyy+j1−22ℓ1+3xux−2ℓ1+1yuy−ℓ1ℓ1+1u=0,y1−yuyy−2xyuxy−2xzuxz+yzuxz−2ℓ2xux+j2−ℓ1+ℓ2+1yuy−ℓ1zuz−ℓ1ℓ2u=0,z1−zuzz−xuxz−yzuyz+ytuyt−ztuzt−ℓ3yuy+j1−ℓ3+ℓ2+1zuz−ℓ2tut−ℓ2ℓ3u=0,t1−tutt−ztuzt−ℓ4zuz+j3−ℓ3+ℓ4+1tut−ℓ3ℓ4u=0.It is noted that four equations of the system (6) are simultaneous. In fact, the hypergeometric function X844 verifies the system. To find the linearly independent solutions of system (6), we will search the solutions in the form
(7)u=xαyβzγtδw,where w is an unknown function and α,β,γ, and δ are constants, which are to be determined. So, substituting u=xαyβzγtδw into the system (6), we get
(8)x1−4xwxx−4xywxy+zwxz−y2wyy+j1+γ+2α−22l1+2α+β+3xwx−2ℓ1+2α+β+1ywy+αx−1zwz−−αj1+α+γ−1x−1+ℓ1+2α+βℓ1+2α+β+1w=0,y1−ywyy−2xywxy−2xzwxz−yzwyz−2ℓ2+α+βxwx+j2+2β−ℓ1+2α+β+ℓ2+β+γ+1ywy−ℓ1+2α+βzwz−−βj2+β−1y−1+ℓ1+2α+βℓ2+β+γw=0,z1−zwzz+xwxz−yzwyz−ytwyt−ztwzt+γxz−1wx−ℓ3+γ+δywy+j1+α+2γ−ℓ2+β+γ+ℓ3+γ+δ+1zwz−ℓ2+β+γtwt−−γj1+α+γ−1z−1+ℓ2+β+γℓ3+γ+δw=0,t1−twtt−ztwzt−ℓ4+δzwz+j3+2δ−ℓ3+γ+δ+ℓ4+δ+1twt−−δj3+δ−1t1+ℓ3+γ+δℓ4+δw=0.Systems (8) and (6) have the same structure and can therefore be approached with similar techniques. System (8) implies
(9)α=0,ββ+j2−1=0,γ=0,δδ+j3−1=0.Therefore, system (9) has the following solutions:
(10)1234α≔0000β≔01−j201−j2γ≔0000δ≔001−j31−j3.Finally, substituting two solutions of the system (10) into (8), we find the following linearly independent solutions of the system (6) at the origin:
(11)u1x,y,z,t=X844ℓ1,ℓ2,ℓ3,ℓ4;j1,j2,j3;x,y,z,t,u2x,y,z,t=y1−j2X844ℓ1+1−j2,ℓ2+1−j2,ℓ3,ℓ4;j1,2−j2,j3;x,y,z,t,u3x,y,z,t=t1−j3X844ℓ1,ℓ2,ℓ3+1−j3,ℓ4+1−j3;j1,j2,2−j3;x,y,z,t,u4x,y,z,t=y1−j2t1−j3X844ℓ1+1−j2,ℓ2+1−j2,ℓ3+1−j3,ℓ4+1−j3;j1,2−j2,2−j3;x,y,z,t.


## 3. Integral Representations of Euler Type

Here, we give eight integral representations of Euler type forX844 whose kernel contains the Gaussian hypergeometric function 2F1 (see [16]), Appell function F3 (see for details [16, 25]), the Exton triple functions X16,X17,X19 [26], Lauricella’s function of three variables FN [16], and the quadruple functions X44,X244 (see [20, 21]):
(12)X844ℓ1,ℓ2,ℓ3,ℓ4;j1,j2,j3;x,y,z,t=Γj2Γj3Γℓ2Γℓ4Γj2−ℓ2Γj3−ℓ4∫0∞∫0∞e−αℓ21−e−αj2−ℓ2−1e−βℓ4×1−e−βj3−ℓ4−11−ye−α−ℓ11−te−β−ℓ3×F3ℓ12,1+ℓ2−j2,ℓ1+12,ℓ3;j1;4x1−ye−α2,−ze−α1−e−α1−te−βdαdβRℓ2>0,Rℓ4>0,Rj2−ℓ2>0,Rj3−ℓ4>0,(13)X844ℓ1,ℓ2,ℓ3,ℓ4;j1,j2,j3;x,y,z,t=2Mℓ2Γj2Γℓ2Γj2−ℓ2∫0∞coshαsinh2αℓ2−1/21+Msinh2αj2−ℓ1×1+Msinh2α−Mysinh2α−ℓ1FNℓ4,ℓ12,1+ℓ2−j2,ℓ3,ℓ1+12,ℓ3;j3,j1,j1;t,4x1+Msinh2α21+Msinh2α−Mysinh2α2,−zMsinh2αdαRℓ2>0,Rj2−ℓ2>0,M>0,(14)X844ℓ1,ℓ2,ℓ3,ℓ4;j1,j2,j3;x,y,z,t=21+Mℓ1Γj2Γℓ1Γj2−ℓ1∫0π/2sin2αℓ1−1/2cos2αj2−ℓ1−1/21+Msin2αj2−ℓ2×1+Msin2α−1+Mysin2α−ℓ2FNℓ4,1+ℓ1−j22,ℓ2,ℓ3,ℓ1−j22+1,ℓ3;j3,j1,j1;t,4x1+M2tan4α,z1+Msin2α1+Msin2α−1+Mysin2αdαRℓ1>0,Rj2−ℓ1>0,M>−1,(15)X844ℓ1,ℓ2,ℓ3,ℓ4;j1,j2,j3;x,y,z,t=2Mj3−ℓ3Γj3Γℓ3Γj3−ℓ3∫0π/2sin2αj3−ℓ3−1/2cos2αℓ3−1/2cos2α+Msin2αj3−ℓ4×cos2α+Msin2α−tcos2α−ℓ4X16ℓ1,ℓ2,1+ℓ3−j3;j1,j2;x,y,−zcot2αMdαRℓ3>0,Rj3−ℓ3>0,M>−1,(16)X844ℓ1,ℓ2,ℓ3,ℓ4;j1,j2,j3;x,y,z,t=Γℓ3+ℓ4Γj2Γℓ1Γℓ3Γℓ4Γj2−ℓ1W1−V1j2−ℓ2−1W2−V2ℓ3+ℓ4−1∫V1W1∫V2W2×α−V1ℓ1−1W1−αj2−ℓ1−1β−V2ℓ3−1W2−βℓ4−1×W1−V1−α−V1y−ℓ2X19ℓ3+ℓ4,ℓ2,1+ℓ1−j22,ℓ1−j22+1;j3,j1;β−V2W2−βtW2−V22,β−V2zW2−V2,4α−V1W1−α2xdαdβRℓ1>0,Rℓ3>0,Rℓ4>0,Rj2−ℓ1>0,V1<W1,V2<W2,(17)X844ℓ1,ℓ2,ℓ3,ℓ4;j1,j2,j3;x,y,z,t=Γℓ3+bΓj12ℓ3+b+j1−2Γℓ3ΓaΓbΓj1−a∫−11∫−111+αℓ3−11−αb−1×1+βa−11−βj1−a−1X17ℓ1,ℓ2,ℓ3+b;a,j2,j1−a;1+βx2,y,1+α1−βz42F1ℓ4,1−b;j3,1+αα−1tdαdβRℓ3>0,Ra>0,Rb>0,Rj1−a>0,(18)X844ℓ1,ℓ2,ℓ3,ℓ4;j1,j2,j3;x,y,z,t=4Γℓ1+ℓ4Γℓ2+ℓ3Γℓ1Γℓ2Γℓ3Γℓ4∫0π/2∫0π/2sin2αℓ1−1/2cos2αℓ4−1/2sin2βℓ2−1/2×cos2βℓ3−1/2X44ℓ1,ℓ1,ℓ2,ℓ1,ℓ1,ℓ2,ℓ2,ℓ2;j1,j2,j1,j3;xsin4α,ysin2αsin2β,zsin22β4,tcos2αcos2βdαdβRℓi>0i=1,2,3,4,(19)X844ℓ1,ℓ2,ℓ3,ℓ4;j1,j2,j3;x,y,z,t=1+Mℓ1Γℓ1+ℓ4Γℓ1Γℓ4∫01αℓ1−11−αℓ4−11+Mαℓ1+ℓ4×X244ℓ1,ℓ1,ℓ2,ℓ1,ℓ1,ℓ2,ℓ3,ℓ4;j1,j2,j1,j3;1+M2α2x1+Mα2,1+Mαy1+Mα,z,1−αt1+MαdαRℓ1>0,Rℓ4>0,M>−1,where the Gaussian hypergeometric function 2F1, Appell function F3, Lauricella triple hypergeometric function FN, Exton hypergeometric functions X16,X17,X19, and the quadruple functions X44,X244 are defined, respectively, by
(20)2F1a,b;c;x=∑n=0∞anbncnxnn!∣x∣<1,(21)F3a,b,c,d;e;x,y=∑m,n=0∞ambncmdnem+nxmm!ynn!max∣x∣,∣y∣<1,(22)FNℓ1,ℓ2,ℓ3,b1,b2,b1;j1,j2,j2;x,y,z=∑m,n,p=0∞ℓ1mℓ2nℓ3pb1m+pb2nj1mj2n+pxmm!ynn!zppr+s<1∧v<1,∣x∣≤r,∣y∣≤s,∣z∣≤v,(23)X16ℓ1,ℓ2,ℓ3;j1,j2;x,y,z=∑m,n,p=0∞ℓ12m+nℓ2n+pℓ3pj1m+pj2nxmm!ynn!zpp!s<1∧v≤Tvs∧r<141−s2∪s<1∧Tvs<v<1−s∧r<v1−vℓ2s1−v,(24)Tvs=1−s1−1/2s2,0<s<23,1−s2s,23≤s<1∣x∣≤r,∣y∣≤s,∣z∣≤v,(25)X17ℓ1,ℓ2,ℓ3;j1,j2,j3;x,y,z=∑m,n,p=0∞ℓ12m+nℓ2n+pℓ3pj1mj2nj3pxmm!ynn!zpp!r<14∧v<1∧s<1−2r1−v,∣x∣≤r,∣y∣≤s,∣z∣≤v,(26)X19ℓ1,ℓ2,ℓ3,ℓ4;j1,j2;x,y,z=∑m,n,p=0∞ℓ12m+nℓ2nℓ3pℓ4pj1mj2n+pxmm!ynn!zpp!s+2r<1∧v<1,∣x∣≤r,∣y∣≤s,∣z∣≤v,(27)X44ℓ1,ℓ1,ℓ2,ℓ1,ℓ1,ℓ2,ℓ2,ℓ2;j1,j2,j1,j3;x,y,z,t=∑m,n,p,q=0∞ℓ12m+n+qℓ2q+n+2pj1m+pj2nj3qxmm!ynn!zpp!tqq!∣x∣<14,∣y∣<1,∣z∣<14,∣t∣<1,(28)X244ℓ1,ℓ1,ℓ2,ℓ1,ℓ1,ℓ2,ℓ3,ℓ3;j1,j2,j1,j3;x,y,z,t=∑m,n,p,q=0∞ℓ12m+n+qℓ2n+pℓ3p+qj1m+pj2nj3qxmm!ynn!zpp!tqq!∣x∣<14,∣y∣<1,∣z∣<1,∣t∣<1.Proof.
We begin by recalling the following integral representations of the beta function (see, for example, [27, 28]):
(29)Ba,b=∫01αα−11−αb−1dαRa>0,Rb>0,Γa+bΓaΓba,bℂ/Z0−,(30)Ba,b=∫01αa−11−αb−1dα=W−V1−a−b∫VWα−Va−1W−αb−1dαRa>0,Rb>0,V<W,(31)Ba,b=2∫0π/2sinα2a−1cosα2b−1dα=∫0∞e−αa1−e−αb−1dαRa>0,Rb>0,(32)Ba,b=21−a−b∫−111+αa−11−αb−1dα=2Ma∫0∞coshαsinhα2a−11+Msinh2αa+bdαRa>0,Rb>0,M>0.
For convenience, let℧ denote the right-hand side of relation (12). Then, by substituting the expression of F3 from definition (21) into the right-hand side of (12) and using (31), we have
(33)℧=∑m,n,p,q=0∞ℓ12mℓ1+2mn1+ℓ2−j2pℓ3pℓ3+pq−1pj1m+p×Γj2Γℓ2Γj2−ℓ2∫0∞e−αℓ2+n+p1−e−αj2−ℓ2−p−1dα×Γj3Γℓ4Γj3−ℓ4∫0∞e−βℓ4+q1−e−βj3−ℓ4−1dβ×xmm!ynn!zpp!tqq!=∑m,n,p,q=0∞1+ℓ2−j2pΓj2−ℓ2−p−1pΓj2−ℓ2×ℓ12m+nℓ2n+pℓ3p+qℓ4qj1m+pj2nj3qxmm!ynn!zpp!tqq!=∑m,n,p,q=0∞ℓ12m+nℓ2n+pℓ3p+qℓ4qj1m+pj2nj3qxmm!ynn!zpp!tqq!=X844ℓ1,ℓ2,ℓ3,ℓ4;j1,j2,j3;x,y,z,t;we are led to the desired result. A similar argument in the proof of relation (12) will be able to establish the results (13)–(19). So, details of the proof are omitted.☐



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