# Generating Functions for Some Hypergeometric Functions of Four Variables via Laplace Integral Representations

Authors: Jihad Younis; Maged Bin-Saad; Ashish Verma
Journal: Journal of Function Spaces
 (2021)

---

## Abstract

Generating functions plays an essential role in the investigation of several useful properties of the sequences which they generate. In this paper, we establish certain generating relations, involving some quadruple hypergeometric functions introduced by Bin-Saad and Younis. Some interesting special cases of our main results are also considered.


---

## Body

## 1. Introduction

The hypergeometric series is the most useful and important special function, and it has been studied to solve various problems in many areas of mathematics, physics, statistics, and engineering [1–5]. Hypergeometric series in several variables appear in numerous fields of applied mathematics, mathematical physics, and chemistry. Very recently, Bin-Saad and Younis [6] introduced thirty new hypergeometric functions of four variables Xi4i=1,2,⋯,30, eight of them are defined below(1)X114a1,a1,a1,a2,a1,a2,a3,a3;c1,c1,c2,c3;x,y,z,u=∑m,n,p,q=0∞a12m+n+pa2n+qa3p+qc1m+nc2pc3qxmm!ynn!zpp!uqq!;(2)X124a1,a1,a1,a2,a1,a2,a3,a3;c2,c1,c1,c3;x,y,z,u=∑m,n,p,q=0∞a12m+n+pa2n+qa3p+qc1n+pc2mc3qxmm!ynn!zpp!uqq!;(3)X164a1,a1,a1,a1,a1,a2,a3,a2;c1,c1,c2,c3;x,y,z,u=∑m,n,p,q=0∞a12m+n+p+qa2n+qa3pc1m+nc2pc3qxmm!ynn!zpp!uqq!;(4)X174a1,a1,a1,a1,a1,a2,a3,a2;c2,c1,c1,c3;x,y,z,u=∑m,n,p,q=0∞a12m+n+p+qa2n+qa3pc1n+pc2mc3qxmm!ynn!zpp!uqq!;(5)X214a1,a1,a2,a1,a1,a2,a3,a2;c1,c1,c2,c3;x,y,z,u=∑m,n,p,q=0∞a12m+n+qa2n+p+qa3pc1m+nc2pc3qxmm!ynn!zpp!uqq!;(6)X224a1,a1,a2,a1,a1,a2,a3,a2;c2,c1,c1,c3;x,y,z,u=∑m,n,p,q=0∞a12m+n+qa2n+p+qa3pc1n+pc2mc3qxmm!ynn!zpp!uqq!;(7)X274a1,a1,a3,a1,a1,a2,a4,a4;c2,c1,c1,c3;x,y,z,u=∑m,n,p,q=0∞a12m+n+qa2na3pa4p+qc1n+pc2mc3qxmm!ynn!zpp!uqq!;(8)X284a1,a1,a1,a1,a1,a2,a3,a4;c1,c1,c2,c3;x,y,z,u=∑m,n,p,q=0∞a12m+n+p+qa2na3pa4qc1m+nc2pc3qxmm!ynn!zpp!uqq!;for(9)x<14,y<1,z<1,u<1.Here,an corresponds to the Pochhammer symbol given as(10)an=Γa+nΓa=aa+1a+2⋯a+n−1,n∈ℕ,and a0=1. Given the following integral representations:(11)X114a1,a1,a1,a2,a1,a2,a3,a3;c1,c1,c2,c3;x,y,z,u=1Γa11Γa2∫0∞∫0∞e−s+tsa1−1ta2−1×0F1−;c1;s2x+styΨ2a3;c2,c3;sz,tudsdt,Ra1>0,Ra2>0;(12)X124a1,a1,a1,a2,a1,a2,a3,a3;c2,c1,c1,c3;x,y,z,u=1Γa21Γa3∫0∞∫0∞e−s+tsa2−1ta3−1×H7a1;c2,c1;x,sy+tz0F1−;c3;studsdt,Ra2>0,Ra3>0;(13)X164a1,a1,a1,a1,a1,a2,a3,a2;c1,c1,c2,c3;x,y,z,u=1Γa11Γa2∫0∞∫0∞e−s+tsa1−1ta2−1×0F1−;c1;s2x+sty1F1a3;c2;sz0F1−;c3;studsdt,Ra1>0,Ra2>0;(14)X174a1,a1,a1,a1,a1,a2,a3,a2;c2,c1,c1,c3;x,y,z,u=1Γa11Γa2∫0∞∫0∞e−s+tsa1−1ta2−1×Φ3a3;c1;sz,sty0F1−;c2;s2x0F1−;c3;studsdt,Ra1>0,Ra2>0;(15)X214a1,a1,a2,a1,a1,a2,a3,a2;c1,c1,c2,c3;x,y,z,u=1Γa11Γa2∫0∞∫0∞e−s+tsa1−1ta2−1×0F1−;c1;s2x+sty1F1a3;c2;tz0F1−;c3;studsdt,Ra1>0,Ra2>0;(16)X224a1,a1,a2,a1,a1,a2,a3,a2;c2,c1,c1,c3;x,y,z,u=1Γa11Γa3∫0∞∫0∞e−s+tsa1−1ta3−1×Ψ2a2;c1,c3;sy+tz,su0F1−;c2;s2xdsdt,Ra1>0,Ra3>0;(17)X274a1,a1,a3,a1,a1,a2,a4,a4;c2,c1,c1,c3;x,y,z,u=1Γa11Γa4∫0∞∫0∞e−s+tsa1−1ta4−1×Φ2a2,a3;c1;sy,tz0F1−;c2;s2x0F1−;c3;studsdt,Ra1>0,Ra4>0;(18)X284a1,a1,a1,a1,a1,a2,a3,a4;c1,c1,c2,c3;x,y,z,u=1Γa1∫0∞e−ssa1−1×Φ3a2;c1;sy,s2x1F1a3;c2;sz1F1a4;c3;suds,Ra1>0,where 0F1,1F1 are Kummer’s functions and Φ2,Φ3,Ψ2,H7 are Humbert functions defined, respectively, by (see [7])(19)0F1−;b;x=∑n=0∞1bnxnn!;(20)F11a;b;x=∑n=0∞anbnxnn!;(21)Φ2a,b;c;x,y=∑m,n=0∞ambncm+nxmm!ynn!;(22)Φ3b;c;x,y=∑m,n=0∞bmcm+nxmm!ynn!;(23)Ψ2a;b,c;x,y=∑m,n=0∞am+nbmcnxmm!ynn!;(24)H7a;b,c;x,y=∑m,n=0∞a2m+nbmcnxmm!ynn!.Several families of generating functions have been established in diverse ways. These are playing important roles in the theory of special functions of applied mathematics and mathematical physics. One can refer to the extensive work of Srivastava and Manocha [8] for a systematic introduction and to several interesting and useful applications of the various methods of obtaining linear, bilinear, bilateral, or mixed multilateral generating functions for a fairly wide variety of sequences of hypergeometric functions and polynomials in one, two, or more variables, among much abundant literature. Many authors have been presented various generating functions in many different ways (see, for details, [9–11] and the references cited therein). In this paper, we aim at establishing some generating functions for the quadruple functions Xi4i=11,12,16,17,21,22,27,28.


## 2. Generating Relations

Here, by using the integral representations in (11)–(18), we give certain generating relations involving hypergeometric functions of three and four variables as follows:(25)∑k=0∞wkk!X114a1+k,a1+k,a1+k,a2+k,a1+k,a2+k,c2,c2;c1,c1,c2,c2;x,y,z,u=1−z−a11−u−a2∑k=0∞1k!w1−z1−uk×X3a1+k,a2+k;c1,c2;x1−z2,y1−z1−u,zu1−z1−u;(26)∑k=0∞wkk!X124a1,a1,a1,a2+k,a1,a2+k,a3+k,a3+k;c2,c1,c1,c3;x,y,z,uτ=1+u−a21+τ−a3∑k,m=0∞a12mc2mk!m!w1+u1+τkxm×F284a1+2m,a1+2m,c3,c3,a2+k,a3+k,a2+k,a3+k;c1,c1,c3,c3;λ1y,λ2z,λ1u,λ2τ,λ1=11+u,λ2=11+τ;(27)∑k=0∞wkk!X164a1+k,a1+k,a1+k,a1+k,a1+k,a2+k,a3,a2+k;c1,c1,c2,c3;x,y,z,u=1−z−a1∑k=0∞1k!w1−zk×X164a1+k,a1+k,a1+k,a1+k,a1+k,a2+k,c2−a3,a2+k;c1,c1,c2,c3;λ2x,λy−λz,λu,λ=11−z;(28)∑k=0∞wkk!X174a1+k,a1+k,a1+k,a1+k,a1+k,a2+k,a3,a2+k;c2,c1,c1,c3;x2,y,z,u=1+2x−a1∑k=0∞1k!w1+2xk×K8a1+k,a1+k,a1+k,a1+k,a2+k,a2+k,a3,c2−12;c1,c3,c1,2c2−1;λy,λu,λz,4λx,λ=11+2x;(29)∑k=0∞wkk!X214a1+k,a1+k,a2+k,a1+k,a1+k,a2+k,a3,a2+k;c1,c1,c2,c3;x,y,z,uτ=1+u−a11+τ−z−a2∑k,p=0∞a2+kpc2−a3pc2pk!p!w1+u1+τ−zkzz−τ−1p×X114a1+,a1+k,a1+k,a2+k+p,a1+k,a2+k+p,c3,c3;c1,c1,c3,c3;λ12x,λ2y,λ1u,λ3τ,λ1=11+u,λ2=11+u1+τ−z,λ3=11+τ−z;(30)∑k=0∞wkk!X224a1+k,a1+k,a2,a1+k,a1+k,a2,a3+k,a2;c2,c1,c1,c3;x2,y,z,u=1+2x−a1∑k,q=0∞a1+kqa2qc3qk!q!w1+2xku1+2xq×FMc2−12,a2+q,a2+q,a1+k+q,a3+k,a1+k+q;2c2−1,c1,c1;4x1+2x,z,y1+2x;(31)∑k=0∞wkk!X274a1+k,a1+k,a3,a1+k,a1+k,a2,a4+k,a4+k;c2,c1,c1,c3;x2,y,z,u=1+2x−a1∑k,q=0∞a1+kqa4+kqc3qk!q!w1+2xku1+2xq×FNc2−12,a3,a2,a1+k+q,a4+k+q,a1+k+q;2c2−1,c1,c1;4x1+2x,z,y1+2x;(32)∑k=0∞wkk!X284a1+k,a1+k,a1+k,a1+k,a1+k,a2,a3,a4;c1,c1,c2,c3;x,y,z,u=1−z−u−a1∑k,n=0∞a1+kna2nc1nk!n!w1−z−uky1−z−un×X8a1+k+n,c2−a3,c3−a4;c1+n,c2,c3;x1−z−u2,zz+u−1,uz+u−1;where X3, X8 are the Exton functions of three variables [12], Lauricella functions FM, FN [13], Exton function of four variables K8 [14], and Sharma and Parihar function F284 [15] are defined, respectively, by(33)X3a1,a2;c1,c2;x,y,z=∑m,n,p=0∞a12m+n+pa2n+pc1m+nc2pxmm!ynn!zpp!;(34)X8a1,a2,a3;c1,c2,c3;x,y,z=∑m,n,p=0∞a12m+n+pa2na3pc1mc2nc3pxmm!ynn!zpp!;(35)FMa1,a2,a2,b1,b2,b1;c1,c2,c2;x,y,z=∑m,n,p=0∞a1ma2n+pb1m+pb2nc1mc2n+pxmm!ynn!zpp!;(36)FNa1,a2,a3,b1,b2,b1;c1,c2,c2;x,y,z=∑m,n,p=0∞a1ma2na3pb1m+pb2nc1mc2n+pxmm!ynn!zpp!;(37)K8a1,a1,a1,a1,b1,b1,b2,b3;c1,c2,c1,c3;x,y,z=∑m,n,p,q=0∞a1m+n+p+qb1m+nb2pb3qc1m+pc2nc3qxmm!ynn!zpp!uqq!;(38)F284a1,a1,a2,a2,b1,b2,b1,b2;c1,c1,c2,c3;x,y,z=∑m,n,p,q=0∞a1m+na2p+qb1m+pb2n+qc1m+nc2pc3qxmm!ynn!zpp!uqq!.Proof.
To prove the above equations, we require the following results (see, e.g., [7, 16, 17]):(39)∑m=0∞∑n=0∞fm+nxmm!ynn!=∑N=0∞fNx+yNN!;(40)Γz=sz∫0∞e−sttz−1dt,Rz>0;(41)an=Γa+nΓa,a≠0,−1,−2,⋯;(42)am+n=ama+mn;(43)0F1−;a;x2=e−2x1F1a−12;2a−1;4x;(44)1F1a;b;x=ex1F1b−a;b;−x;(45)Ψ2c,c;c,x,y=expx+y0F1−;c;xy.
For convenience and simplicity, by denoting the left-hand side of (25) by δ, and using (11), one gets(46)δ=∑k=0∞wkk!Γa1+kΓa2+k∫0∞∫0∞sa1+k−1ta2+k−1×F01−;c1;s2x+styΨ2c2;c2,c2;sz,tudsdt.
In view of (40) and (46), we have(47)δ=∑k,m,n=0∞wkxmync1m+nk!m!n!Γa1+kΓa2+k∫0∞∫0∞e−s1−ze−t1−u×sa1+k+2m+n−1ta2+k+n−1F01−;c2;stzudsdt.
The function0F1 which appears in the above equation can be replaced by its series form and then interchanging the order of the summation and integral sign which is permissible here, we get(48)δ=∑k,m,n,p=0∞wkxmynzupc1m+nc2pk!m!n!p!Γa1+kΓa2+k×∫0∞∫0∞e−s1−ze−t1−usa1+k+2m+n+p−1ta2+k+n+p−1dsdt.
Now, the use of (41) and (42) in above equation and the simplification with series manipulation completes the proof of relation (25).
The proof of all remaining relations runs in the same way, considering the appropriate integral representation and Laplace transform during the proof.☐


## 3. Special Cases

Here, we shall consider several interesting special cases of our main results stated in the previous section.Puttingk=0 in equations (25) to (28), we obtain the following relations:(49)X114a1,a1,a1,a2,a1,a2,c2,c2;c1,c1,c2,c2;x,y,z,u=1−z−a11−u−a2×X3a1,a2;c1,c2;x1−z2;y1−z1−u,zu1−z1−u;(50)X124a1,a1,a1,a2,a1,a2,a3,a3;c2,c1,c1,c3;x,y,z,uτ=1+u−a21+τ−a2∑m=0∞a12mc2mm!xm×F284a1+2m,a1+2m,c3,c3,a2,a3,a2,a3;c1,c1,c3,c3;y1+u,z1+τ,u1+u,τ1+τ;(51)X164a1,a1,a1,a1,a1,a2,a3,a2;c1,c1,c2,c3;x,y,z,u=1−z−a1X164a1,a1,a1,a1,a1,a2,c2,−a3,a2;c1,c1,c2,c3;x1−z2,y1−z,zz−1,u1−z;(52)X174a1,a1,a1,a1,a1,a2,a3,a2;c2,c1,c1,c3;x2,y,z,u=1+2x−a1×K8a1,a1,a1,a1,a2,a2,a3,c2−12;c1,c3,c1,2c2−1;λy,λu,λz,4λx,λ=11+2x.Equations (49), (51), and (52) with u=0 yield the Exton’s results (see [12]).Now, if we takex=0 in (25) to (27) and (29), and simplification, we shall obtain the following generating relations:(53)∑k=0∞wkk!HBa1+k,a2+k,c2;c1,c3,c2;y,u,z=1−z−a11−u−a2∑k=0∞1k!w1−z1−uk×F4a1+k,a2+k;c1,c2;y1−z1−u,zu1−z1−u;(54)∑k=0∞wkk!HAa2+k,a3+k,a1;c3,c1;uτ,y,z=1+u−a21+τ−a3∑k=0∞1k!w1+τ1+uk×F284a1,a1,c3,c3,a2+k,a3+k,a2+k,a3+k;c1,c1,c3,c3;y1+u,z1+τ,u1+u,τ1+τ;(55)∑k=0∞wkk!FEa1+k,a1+k,a1+k,a3,a2+k,a2+k;c2,c1,c3;z,y,u=1−z−a1∑k=0∞1k!w1−zk×FEa1+k,a1+k,a1+k,c2−a3,a2+k,a2+k;c2,c1,c3;zz−1,y1−z,u1−z;(56)∑k=0∞wkk!FEa2+k,a2+k,a2+k,a3,a1+k,a1+k;c2,c1,c3;z,y,uτ=1+u−a11+τ−z−a2∑k,p=0∞a2+kpc2−a3pc2pk!p!w1+u1+τ−zkzz−τ−1p×HBa1+k,a2+k+p,c3;c1,c3,c3;λ1y,λ2τ,uu+1,λ1=11+u1+τ−z,λ2=11+τ−z.Note that the special cases of each of the above generating relations can be easily derived by assigning the value zero tok. For example,(57)HBa1,a2,c2;c1,c2;y,u,z=1−z−a11−u−a2F4a1,a2;c1,c2;y1−z1−u,zu1−z1−u,which, when a2=c3, yields [[7], pp. 309 (125)].Another interesting special case of (56) occurs when we set u=0. We thus find that(58)F2a1,a2,c2;c1,c2;y,z=1−z−a1F21a1,a2;c1;y1−z,which is due to Srivastava and Karlsson [7].Also,(59)HAa2,a3,a1;c3,c1;uτ,y,z=1+u−a21+τ−a3F284a1,a1,c3,c3,a2,a3,a2,a3;c1,c1,c3,c3;y1+u,z1+τ,u1+u,τ1+τ,which, for y=z=0, x=u/1+u and y=τ/1+τ, yields the well-known result [7].Furthermore, by settingk=0 and y=0 in (25) and (32), we obtain the Exton’s results [12].Finally, if in (29), we let z=0, we shall obtain generating relation between Exton’s series X3 and the quadruple hypergeometric series X114:(60)∑k=0∞wkk!X3a1+k,a2+k;c1,c3;x,y,uτ=1+u−a11+τ−a2∑k=0∞1k!w1+u1+τk×X114a1+k,a1+k,a1+k,a2+k,a1+k,a2+k,c3,c3;c1,c1,c3,c3;λ12x,λ1λ2y,λ1u,λ2τ,λ1=11+u,λ2=11+τ.Formula (61), with x=0, yields the generating relation(61)∑k=0∞wkk!F4a1+k,a2+k;c1,c3;y,uτ=1+u−a11+τ−a2∑k=0∞1k!w1+u1+τk×HBa1+k,a2+k,c3;c1,c3,c3;y1+u1+τ,τ1+τ,u1+u.Fork=0, we have the elegant transformation(62)F4a1,a2;c1,c3;y,uτ=1+u−a11+τ−a2HBa1,a2,c3;c1,c3,c3;y1+u1+τ,τ1+τ,u1+u;where 2F1, F2, F4, HA, HB, and FE are the Gaussian hypergeometric function, Appell’s functions, Srivastava’s functions, and Lauricella function defined, respectively, by (see [7])(63)2F1a,b;c;z=∑n=0∞anbncnznn!;(64)F2a,b,c;d,e;x,y=∑m,n=0∞am+nbmcndmenxmm!ynn!;(65)F4a,b;c,d;x,y=∑m,n=0∞am+nbm+ncmdnxmm!ynn!;(66)HAa1,b1,b2;c1,c2;x,y,z=∑m,n,p=0∞a1m+pb1m+nb2n+pc1mc2n+pxmm!ynn!zpp!;(67)HBa1,b1,b2;c1,c2,c3;x,y,z=∑m,n,p=0∞a1m+pb1m+nb2n+pc1mc2nc3pxmm!ynn!zpp!;(68)FEa1,a1,a1,b1,b2,b2;c1,c2,c3;x,y,z=∑m,n,p=0∞a1m+n+pb1mb2n+pc1mc2nc3pxmm!ynn!zpp!.


## 4. Conclusion

Based on the integral representations for quadruple hypergeometric functions (1)–(8), we obtained certain generating functions for these functions. Some particular cases and the consequences of our main results are also considered. We concluded this investigation by remarking that the scheme suggested in the derivation of the results can be applied to find other new generating functions for other quadruple hypergeometric functions and study their special cases.



---

*Source: 7638597-2021-07-15.xml*