# On Gaussian Hypergeometric Functions of Three Variables: Some New Integral Representations

Authors: Maged G. Bin-Saad; Mohannad J. S. Shahwan; Jihad A. Younis; Hassen Aydi; Mohamed A. Abd El Salam
Journal: Journal of Mathematics
 (2022)

---

## Abstract

The present paper establishes several new integral representations of the Euler type and Laplace type for some Gauss hypergeometric functions of three variables. The main results are obtained by using the properties of Gamma and beta functions. The novel integral representations are carried out through ten hypergeometric functions of three variables. Therefore, all derived integrals are generalization representation of the Euler type for the classical Gauss hypergeometric function of one and two variables. In addition, several numerical examples were given to describe some of the obtained results.


---

## Body

## 1. Introduction

Special functions are particular mathematical functions with more or less established names and notations due to their significance in mathematical analysis. The hypergeometric function is one of the essential special functions that have had more attention in recent years, including many other special functions as specific or limiting cases. The hypergeometric functions and their generalized extensions have been introduced and investigated extensively due mainly to their applications in diverse areas in mathematics, applied mathematics, physics, and engineering. Srivastava and Karlsson[1] introduced and analyzed the extension of the generalized hypergeometric function utilizing the extended Pochhammer symbol. Another extension of the generalized hypergeometric functions is defined in reference [2]. Ref. [3] (Ali et al. introduced the multiindex Gauss hypergeometric functions using the multiindex Beta function. Saboor et al. [4] characterized a new extension of Srivastava’s triple hypergeometric functions; besides, they presented their specific properties, derivative formulas, integral representations, and some new recurrence relations. Recently, numerous researchers have studied several extensions and generalizations of many special functions (see, e.g., [5–11]).It is known that there are 205 hypergeometric functions of three variables of second order, of which regions of convergence have been given in the literature. Hypergeometric functions of three variables can be expressed either by integrals of the Euler type or by integrals of the Laplace type [12]. The list of hypergeometric functions of three variables is too extensive (see [13], Chapter 3), and it is impossible to give a complete list of integral representations here. Integral representations of the hypergeometric functions with three variables are helpful for the analytical continuation [14]. In addition, an analytic continuation of the Horn hypergeometric function with an arbitrary number of variables is given in [15]. Therefore, the integral representations are mainly in the theory of transformation, as well as the integration of hypergeometric systems of partial differential equations (PDEs) (see [5]). In the attending work, the authors aim to obtain some new integral representations for hypergeometric functions of three variables by applying either the first-kind Euler integral or the integral of the second kind.


## 2. Preliminaries

In this section, the definition and some helpful relations will concern. For our purpose, we begin by recalling some Gauss hypergeometric functions [13] of three variables as follows:(1)F4da1,a2,a3,a4,b;c1,c2;x,y,z=∑m,n,p=0∞a1m+na2ma3na4pbp−mc1nc2pxmm!ynn!zpp!,(2)F8ca1,a2,a3,b;c1,c2;x,y,z=∑m,n,p=0∞a1m+na2m+na3pbp−mc1nc2pxmm!ynn!zpp!,(3)F10da1,a2,a3,b;c1,c2;x,y,z=∑m,n,p=0∞a1m+na2n+pa3pbm−nc1mc2pxmm!ynn!zpp!,(4)F17ca1,a2,b1,b2;c1;x,y,z=∑m,n,p=0∞a1m+n+pa2pb1m−nb2n−mc1pxmm!ynn!zpp!,(5)F26ba1,a2,a3,b1;c1,c2;x,y,z=∑m,n,p=0∞a1m+na2ma3nb12p−mc1nc2pxmm!ynn!zpp!,(6)F26da1,a2,b1,b2;c;x,y,z=∑m,n,p=0∞a1m+na2nb12p−mb2m−pcnxmm!ynn!zpp!,(7)F29ja1,a2,b1,b2;c;x,y,z=∑m,n,p=0∞a12m+na2pb1n−pb2p−ncmxmm!ynn!zpp!,(8)F36ea1,a2,b;c1,c2;x,y,z=∑m,n,p=0∞a12m+na2n+pbp−nc1mc2pxmm!ynn!zpp!,(9)F41ba1,a2,b1;c1,c2;x,y,z=∑m,n,p=0∞a12m+n+pa2pb1n−mc1nc2pxmm!ynn!zpp!,(10)F41ea,b1,b2;c;x,y,z=∑m,n,p=0∞a2m+n+pb1n−pb2p−ncmxmm!ynn!zpp!.We note thatLk is the well-known Pochhammer’s symbol, and it was handed by(11)Lk=ΓL+kΓL=1,k=0,LL+1…L+k−1,k∈ℕ0=1,2,3….The Gauss hypergeometric functionF21 is defined as (see Ref. [12])(12)F21a,b;c;x=∑m=0∞ambmcmxmm!,x<1.The two-variables hypergeometric functionsF2 and F4 are, respectively, defined by (see [16, 17])(13)F2a,b,c;d,e;x,y=∑m,n=0∞am+nbmcndmenxmm!ynn!,and(14)F4a,b;c,d;x,y=∑m,n=0∞am+nbm+ncmdnxmm!ynn!.Horn’s functions of two variables (see Ref. [18]), namely, G1,G2,H1,H2,H4,H5,H6, and H7, are given as(15)G1a,b,c;x,y=∑m,n=0∞am+nbn−mcm−nxmm!ynn!,G2a,b,c,d;x,y=∑m,n=0∞ambncn−mdm−nxmm!ynn!,H1a,b,c;d;x,y=∑m,n=0∞am−nbm+ncndmxmm!ynn!,H2a,b,c,d;e;x,y=∑m,n=0∞am−nbmcndnemxmm!ynn!,H4a,b;c,d;x,y=∑m,n=0∞a2m+nbncmcnxmm!ynn!,H5a,b;c;x,y=∑m,n=0∞a2m+nbn−mcnxmm!ynn!,H6a,b,c;x,y=∑m,n=0∞a2m−nbn−mcnxmm!ynn!,H7a,b,c;d;x,y=∑m,n=0∞a2m−nbncndmxmm!ynn!.The useful Lauricella functions of three variables (see [13, 19]), namely, FE,FK, and FR, are characterized by(16)FEa1,a1,a1,b1,b2,b2;c1,c2,c3;x,y,z=∑m,n,p=0∞a1m+n+pb1mb2n+pc1mc2nc3pxmm!ynn!zpp,FKa1,a2,a2,b1,b2,b1;c1,c2,c3;x,y,z=∑m,n,p=0∞a1ma2n+pb1m+pb2nc1mc2nc3pxmm!ynn!zpp,FRa1,a2,a1,b1,b2,b1;c1,c2,c2;x,y,z=∑m,n,p=0∞a1m+pa2nb1m+pb2nc1mc2n+pxmm!ynn!zpp.Lastly, the three-variable Exton’s functions (see Ref. [20]), namely, X1,X2,X6,X8,X15,X17, and X20, are used here and expressed as follows:(17)X1a,b;c,d;x,y,z=∑m,n,p=0∞a2m+2n+pbpcmdn+pxmm!ynn!zpp!,X2a1,a2;c1,c2,c3;x,y,z=∑m,n,p=0∞a12m+2n+pa2pc1mc2nc3pxmm!ynn!zpp!,X6a1,a2,a3;c1,c2;x,y,z=∑m,n,p=0∞a12m+n+pa2na3pc1m+nc2pxmm!ynn!zpp!,X8a1,a2,a3;c1,c2,c3;x,y,z=∑m,n,p=0∞a12m+n+pa2na3pc1mc2nc3pxmm!ynn!zpp!,X15a1,a2,a3;c2,c1;x,y,z=∑m,n,p=0∞a12m+na2n+pa3pc1n+pc2mxmm!ynn!zpp!,X17a1,a2,a3;c1,c2,c3;x,y,z=∑m,n,p=0∞a12m+na2n+pa3pc1mc2nc3pxmm!ynn!zpp!,X20a1,a2,a3,a4;c1,c2;x,y,z=∑m,n,p=0∞a12m+na2na3pa4pc1m+pc2nxmm!ynn!zpp!.Now, we turn to the main results of the work.


## 3. Integral Representations of Euler Type

In the current section, we present certain Euler-type integral representations of the Gaussian triple hypergeometric functions defined in (1) to (10), whose kernels contain Gauss hypergeometric function F21, Appell’s functions F2 and F4, Horn hypergeometric functions G1,G2,H1,H2,H4,H5,H6, and H7, the Lauricellas triple functions FE,FK, and FR, and the three variables Exton’s functions X1,X2,X6,X8,X15,X17, and X20.Theorem 1.
The following integral representations hold true:(18)F4da1,a2,a3,a4,b;c1,c2;x,y,z=Γc2S−Ta4R−Tc2−a4Γa4Γc2−a4S−Rc2−b−1×∫RSα−Ra4−1S−αc2−a4−1α−Tc2−bS−Rα−T−S−Tα−Rz−b×F2a1,a2,a3;1−b,c1;S−Rα−TS−Tα−Rz−1x,ydα,Rea4>0andRec2−a4>0,T<R<S,(19)F4da1,a2,a3,a4,b;c1,c2;x,y,z=Γc12c1−2Γa3Γc1−a3×∫−111+α2a3−1/21−α2c1−a3−1/21+α2c1−a11+α2−121+α2y−a1×H2b,a4,a1,a2;c2;z,1+α2x1+α2−1/21+α2ydα,Rea3>0andRec1−a3>0,(20)F4da1,a2,a3,a4,b;c1,c2;x,y,z=4Γa3+a4Γb+b′Γa3Γa4ΓbΓb′∫0π/2∫0π/2sin2αa3−1/2×cos2αa4−1/2sin2βb−1/2cos2βb′−1/2FKa2,a3+a4,a3+a4,a1,b+b′,a1;b′,c2,c1;xcos2β,zcos2αsin2β,ysin2αdαdβ,Rea3>0,Rea4>0,Reb>0,Reb′>0.Proof.
To prove the relation (18) confirmed in Theorem 1, let ℵ denotes its right-hand side. Then, from the definition of Appell’s function F2 in (13), we get(21)ℵ=Γc2Γa4Γc2−a4∑m,n,p=0∞a1m+na2ma3nb−mb−mpc1n×S−Ta4+pR−Tc2−a4S−Rc2+p−1∫RSα−Ra4+p−1S−αc2−a4−1α−Tc2+p×xmm!ynn!zpp!dα.
By applying the following integral representation of the beta function (see, e.g., [17], p. 10, (20))(22)Ba,b=S−TaR−TbS−Ra+b−1∫RSα−Ra−1S−αb−1α−Ta+bdα,T<R<S,Rea>0,Reb>0,
in (21), we obtain(23)ℵ=Γc2Γa4Γc2−a4∑m,n,p=0∞a1m+na2ma3nbp−mc1nBa4+p,c2−a4xmm!ynn!zpp!.
Now, using the well-known beta function (see, e.g., [17, 21])(24)Ba,b=ΓaΓbΓa+b,
in (16), we get the required result. Using the same manner, we find the identities (19) and (20).Using a similar demonstration as the previous proof, we can give the following theorems.Theorem 2.
The next integrals are true as representations ofF8c:(25)F8ca1,a2,a3,b;c1,c2;x,y,z=Γc2Γa3Γc2−a3∫0∞αa3−11+αc2−b×1+α−αz−bF4a1,a2;1−b,c1;αz1+α−1x,ydα,Rea3>0,Rec2−a3>0,(26)F8ca1,a2,a3,b;c1,c2;x,y,z=41+M1a11+M2bΓa1+a3Γb+b′Γa1Γa3ΓbΓb′×∫0π/2∫0π/2sin2αa1−1/2cos2αa3−1/21+M1sin2αa1+a3sin2βb−1/2cos2βb′−1/21+M2sin2βb+b′×FEa1+a3,a1+a3,a1+a3,b+b′,a2,a2;c2,b′,c1;1+M2zcos2αsin2β1+M1sin2α1+M2sin2β,1+M1xsin2αcot2β1+M1sin2α1+M2sin2βdαdβ,Rea1>0,Rea3>0,Reb>0,Reb′>0,M1>−1,M2>−1,(27)F8ca1,a2,a3,b;c1,c2;x,y,z=Γa1+a2+a3Γb+b′Γa1Γa2Γa3ΓbΓb′∫01∫01∫01αa1−1×1−αa2−1βb′−11−βb−1γa1+a2−11−γa3−1X2a1+a2+a3,b+b′;b′,c1,c2;α1−αβγ2x1−β,α1−αγ2y,1−β1−γzdαdβdγ,Rea1>0,Rea2>0,Rea3>0,Reb>0,Reb′>0.Theorem 3.
The next integrals are true as representations ofF10d:(28)F10da1,a2,a3,b;c1,c2;x,y,z=Γa1+bΓa1ΓbS−Ra1+b−1∫RSα−Ra1−1×S−αa2+b−1S−α−α−Ry−a2×2F1a1+b2,a1+b+12;c1;4α−RS−αxS−R22F1a2,a3;c2;S−αzS−α−α−Rydα,Rea1>0,Reb>0,R<S,(29)F10da1,a2,a3,b;c1,c2;x,y,z=Γc2Γa3Γc2−a3∫0∞e−αa31−e−αc2−a3−1×1−ze−α−a2H1b,a1,a2;c1;x,y1−ze−αdα,Rea3>0,Rec2−a3>0,(30)F10da1,a2,a3,b;c1,c2;x,y,z=4M1a1M2bΓa1+a2Γb+b′Γa1Γa2ΓbΓb′×∫0∞∫0∞coshαsinh2αa1−1/21+M1sinh2αa1+a2coshβsinh2βb−1/21+M2sinh2βb+b′X8a1+a2,b+b′,a3;b′,c1,c2;M1ysinh2αcsch2βM21+M1sinh2α,M1M2xsinh2αsinh2β1+M1sinh2α1+M2sinh2β,z1+M1sinh2αdαdβ,Rea1>0,Rea2>0,Reb>0,Reb′>0,M1>0,M2>0.Theorem 4.
The next integrals are true as representations ofF17c:(31)F17ca1,a2,b1,b2;c1;x,y,z=Γc1Γa1Γc1−a1∫−1/21/212+αa1−112−αc1−a1−1×1−12+αz−a2G11+a1−c1,b2,b1;1+2α2α−1x,1+2α2α−1ydα,Rea1>0,Rec1−a1>0,(32)F17ca1,a2,b1,b2;c1;x,y,z=1+Ma1Γa1+b1Γa1Γb1∫01αa1−11−αb1+b2−11+Mαa1+b1×1−α−1+Mαy−b2H4a1+b1,a2;1−b2,c1;−1+Mα1−α−1+Mαyx1+Mα2,1+Mαz1+Mαdα,Rea1>0,Reb1>0,M>−1,(33)F17ca1,a2,b1,b2;c1;x,y,z=Γa1+b1Γb2+b′2a1+b1+b2+b′−2Γa1Γb1Γb2Γb′∫−11∫−111+αa1−1×1−αb1−11+βb′−11−βb2−1X20a1+b1,a2,b2+b′2,b2+b′+12;b′,c1;1−α21−βx41+β,1+αz2,1+α1−β2y1−αdαdβ,Rea1>0,Reb1>0,Reb2>0,Reb′>0.Theorem 5.
The next integrals are true as representations ofF26b:(34)F26ba1,a2,a3,b1;c1,c2;x,y,z=Γc22c2−2Γb1Γc2−b1∫−111+α2b1−1/21−α21/21+α22c2−b1−1×1−α21+α2+121+α4zc2−b1−1F2a1,a2,a3;c2−b1,c1;1−α21+α2+1/21+α4zx1+α21+α2,ydα,Reb1>0andRec2−b1>0,(35)F26ba1,a2,a3,b1;c1,c2;x,y,z=2Ma2+b1Γa1+b1Γa1Γb1∫0π/2sin2αb1−1/2cos2αa1−1/2cos2α+Msin2αa1+b1×M−xcot2α−a2H4a1+b1,a3;c2,c1;M2zsin4αcos2α+Msin2α2,ycos2αcos2α+Msin2αdα,Rea1>0,Reb1>0,M>0,(36)F26ba1,a2,a3,b1;c1,c2;x,y,z=Γc2S−Ta3R−Tc1−a3Γa3Γc2−a3S−Rc1−a1−1×∫RSα−Ra3−1S−αc1−a3−1α−Tc1−a1S−Rα−T−S−Tα−Ry−a1×H7b1,a1,a2;c2;z,S−Rα−TxS−Rα−T−S−Tα−Rydα,Rea3>0,Rec2−a3>0,T<R<S.Theorem 6.
The next integrals are true as representations ofF26d:(37)F26da1,a2,b1,b2;c;x,y,z=1+Ma1ΓcΓa1Γc−a1∫01αa1−11−αc−a1−11+Mαc−a2×1+Mα−1+Mαy−a2H6b1,b2,1+a1−c;z,1+Mαxα−1dα,Rea1>0,Rec−a1>0andM>−1,(38)F26da1,a2,b1,b2;c;x,y,z=Γb2+b′ΓcΓa2Γb2Γb′Γc−a2∫0∞∫0∞αa2−11+αc−a1βb2−11+βb2+b′×1+α−αy−a1H7b1,a1,b2+b′;b′;zβ,1+αβx1+β1+α−αydαdβ,Rea2>0,Reb2>0,Reb′>0andRec−a2>0,(39)F26da1,a2,b1,b2;c;x,y,z=8Γa1+a2+b1Γb2+b′Γa1Γa2Γb1Γb2Γb′∫0π/2∫0π/2∫0π/2sin2αb2−1/2×cos2αb′−1/2sin2βa1−1/2cos2βa2−1/2sin2γa1+a2−1/2cos2γb1−1/2×FRa1+a2+b12,b2+b′2,a1+a2+b12,a1+a2+b1+12,b2+b¢+12,a1+a2+b1+12;c2,b′,b′;ysin22βsin4γ,xsin22αsin2βtan2γ,4zcot2αcos4γdαdβdγ,Rea1>0,Rea2>0,Reb1>0,Reb2>0,Reb′>0.Theorem 7.
The next integrals are true as representations ofF29j:(40)F29ja1,a2,b1,b2;c;x,y,z=2Ma2+b1Γb1+b2Γb1Γb2∫0π/2sin2αb1−1/2cos2αb2−1/2cos2α+Msin2αb1+b2×1−Mytan2α−a1M−zcot2α−a22F1a12,a1+12;c;4x1−Mytan2α2dα,Reb1>0,Reb2>0,M>0,(41)F29ja1,a2,b1,b2;c;x,y,z=Γc2c−1Γa1Γc−a1∫−111+αa1−1×1−α+121+α2xc−a1−1G21+a1−c,a2,b2,b1;−1+αy1−α+1/21+α2x,zdα,Rea1>0andRec−a1>0,(42)F29ja1,a2,b1,b2;c;x,y,z=Γa1+a2Γb1+b′Γb2+b″Γa1Γa2Γb1Γb2Γb′Γb″∫0∞∫0∞∫0∞e−αa1×1−e−αa2−1e−βb11−e−βb′−1e−γb21−e−γb″−1×X8a1+a2,b1+b′,b2+b″;c,b″,b′;xe−2α,ye−α+β−γ1−e−γ,zeβ−γ1−e−α1−e−βdαdβdγ,Rea1>0,Rea2>0,Reb1>0,Reb2>0,Reb′>0andReb″>0.Theorem 8.
The next integrals are true as representations ofF36e:(43)F36ea1,a2,b;c1,c2;x,y,z=2MbΓc2ΓbΓc2−b∫0∞coshαsinh2αb−1/21+Msinh2αc2−a2×1+Msinh2α−Mzsinh2α−a2×H4a1,a2;c1,c2−b;x,1+Msinh2αyM1+Msinh2α−Mzsinh2αsinh2αdα,Reb>0,Rec2−b>0andM>0,(44)F36ea1,a2,b;c1,c2;x,y,z=Γ2c2ΓbΓ2c2−b∫−1/21/212+αb−112−α2c2−b−1×X15a1,a2,c+12;c1,2c2−b;x,1−2α1+2αy,1−4α2zdα,Reb>0andRe2c2−b>0,(45)F36ea1,a2,b;c1,c2;x,y,z=Γb+b′S−Tb′R−TbΓbΓb′S−Rb+b′−1∫RSα−Rb′−1S−αb−1α−Tb+b′×X17a1,a2,b+b′;c1,b′,c2;x,S−Tα−RyR−TS−α,R−TS−αzS−Rα−Tdα,Reb>0andReb′>0where,T<R<S.Theorem 9.
The next integrals are true as representations ofF41b:(46)F41ba1,a2,b1;c1,c2;x,y,z=Γc2Γa2Γc2−a2S−Rc2−a1−1∫RSα−Ra2−1S−αc2−a2−1×S−R−α−Rz−a1H5a1,b1;c1;S−R2xS−R−α−Rz2,S−RyS−R−α−Rzdα,Rea2>0,Rec2−a2>0,R<S,(47)F41ba1,a2,b1;c1,c2;x,y,z=1+M1a11+M2b1Γa1+a2Γ2c1Γa1Γa2Γb1Γ2c1−b1∫01∫01αa1−11−αa2−11+M1αa1+a2×βb1−11−β2c1−b1−11+M2β2c1X1a1+a2,c1+12;c2,2c1−b1;1+M1α1−αz1+M1α2,1+M12α21−βx1+M21+M1α2β,41+M11+M2αβ1−βy1+M1α1+M2β2dαdβ,Rea1>0,Rea2>0,Reb1>0,Re2c1−b1>0,M1>−1,M2>−1,(48)F41ba1,a2,b1;c1,c2;x,y,z=Γ2c122c1−1Γb1Γ2c1−b1∫−111+α2b1−1/21−α22c1−b1−1/21+α22c1×X6a1,c1+12,a2;2c1−b1,c2;1−α1+α2x,1−α21+α22y,zdα,Reb1>0,Re2c1−b1>0.Theorem 10.
The next integrals are true as representations ofF41e:(49)F41ea,b1,b2;c;x,y,z=Γa+b12a+b1−1ΓaΓb1∫−111+αa−11−αb1+b2−1×1−α−1+αz−b2×F4a+b12,a+b1+12;c;1−b2;1+α2x,−1+α1−α−1+αzydα,Rea>0andReb1>0,(50)F41ea,b1,b2;c;x,y,z=ΓcΓaΓc−a∫01αa−11−α+α2xc−a−1×G11+a−c,b1,b2;d;−αz1−α+α2x,−αy1−α+α2xdα,Rea>0andRec−a>0,(51)F41ea,b1,b2;c;x,y,z=21+Mb2Γb2+b′Γb2Γb′∫0π/2sin2αb2−1/2cos2αb′−1/21+Msin2αb2+b′×F41ea,b1,1−b′;c;x,ycot2α1+M,1+Mztan2αdα,whereReb2>0andReb′>0,M>−1.Now, we introduce some numerical examples for an integral representation to illustrate the above results. Consider the following integral representation of the Euler type.If we consider the integral representation given in Eq. (29) with particular values of a1,a1,a2,a2;c1,c1;x,y and z, we will get different expressions of F10d. If we set x=y=0 and z=1 in (29), we obtain after simplifications(52)F10da1,a1,a2,a2;c1,c1;0,0,1=Γc1Γa2Γc1−a2,∫01αa2−11−αc1−a1−a2−1dα=Γc1Γa2Γc1−a2Γa2Γc1−a1−a2Γc1−a1=c1−a2−a1c1−a1.Putting a1=1 in (52), we have(53)F10d1,1,a2,a2;c1,c1;0,0,1=1−c11+a2−c1.Its plot is in Figure 1, and in addition, the following examples hold true.Figure 1
Plot of functionga2,c1=F10d1,1,a2,a2;c1,c1;0,0,1 given in equation (45), with c1∈0,1.5 and a2∈0,1 and c1>a2.Example 1.
Fora2=1/2 and c1=1 in (53), we get(54)F10d1,1,12,12;1,1;0,0,1=0.Example 2.
If we leta2=1/6 and c1=1/3 in (57), we have(55)F10d1,1,16,16;13,13;0,0,1=45.Example 3.
By takinga2=2 and c1=5 in (57), we get(56)F10d1,1,2,2;5,5;0,0,1=2.


## 4. Integral Representations of Laplace Type

In the next, we present Laplace-type integrals for each function from (1) to (10) by means of the confluent hypergeometric functions 0F1,1F1 and Humbert functions Γ1,Γ2,H1,H3,H5,H8, and H9. First of all, we recall the following confluent hypergeometric functions (see [17]):(57)0F1−;c;x=∑m=0∞1cmxmm!,1F1a;c;x=∑m=0∞amcmxmm!,Γ1a,b;x,y=∑m,n=0∞ambn−mcm−nxmm!ynn!,Γ2a,b,c;x,y=∑m,n=0∞an−mbm−nxmm!ynn!,(58)H1a,b;c;x,y=∑m,n=0∞am−nbm+ncmxmm!ynn!,H3a,b;c;x,y=∑m,n=0∞am−nbmcmxmm!ynn!,H5a;b;x,y=∑m,n=0∞am−nbmxmm!ynn!,H8a,b;x,y=∑m,n=0∞a2m−nbn−mxmm!ynn!,H9a,b;c;x,y=∑m,n=0∞a2m−nbncmxmm!ynn!.Theorem 11.
For the hypergeometric functionsF4d,F8c,F10d,F17c,F26b,F26d,F29j,F36e,F41bandF41e, we obtain(59)F4da1,a2,a3,a4,b;c1,c2;x,y,z=1Γa1Γa2∫0∞∫0∞e−s+tsa1−1ta2−1×1F1a3;c1;syH3b,a4;c2;z,stxdsdt,whereRea1>0andRea2>0,(60)F8ca1,a2,a3,b;c1,c2;x,y,z=1Γa1Γa2∫0∞∫0∞e−s+tsa1−1ta2−1×0F1−;c1;styH3b,a3;c2;z,stxdsdt,whereRea1>0andRea2>0,(61)F10da1,a2,a3,b;c1,c2;x,y,z=1Γa2∫0∞e−ssa2−1H1b,a1;c1;x,sy1F1a3;c2;szds,Rea2>0,(62)F17ca1,a2,b1,b2;c1;x,y,z=1Γa1∫0∞e−ssa1−1Γ2b2,b1;sx,sy1F1a2;c1;szds,Rea1>0,(63)F26ba1,a2,a3,b1;c1,c2;x,y,z=1Γa1∫0∞e−ssa1−11F1a3;c1;syH9b1,a2;c2;z,sxds,Rea1>0,(64)F26da1,a2,b1,b2;c;x,y,z=1Γa1∫0∞e−ssa1−11F1a2;c;syH8b1,b2;z,sxds,Rea1>0,(65)F29ja1,a2,b1,b2;c;x,y,z=1Γa1∫0∞e−ssa1−10F1−;c;s2xΓ1a2,b1,b2;z,syds,Rea1>0,(66)F36ea1,a2,b;c1,c2;x,y,z=1Γa1∫0∞e−ssa1−10F1−;c1;s2xH1b,a2;c2;z,syds,Rea1>0,(67)F41ba1,a2,b1;c1,c2;x,y,z=1Γa1∫0∞e−ssa1−11F1a2;c2;szH5b1;c1;sy,s2xds,Rea1>0,(68)F41ea,b1,b2;c;x,y,z=1Γa∫0∞e−ssa−10F1−;c;s2xΓ2b2,b1;sy,szds,Rea>0.Proof.
The previous integral representations from Eq. (58) to Eq. (67) were proved by substituting the series definition (57) which involved the confluent hypergeometric functions in each integral and by using the change of order of the integrals sign and the summations, in addition, by the help of the following known gamma integral formula [21]:(69)Γx=∫0∞e−ttx−1dt,whereRex>0.
Consider the following integral representation of the Laplace type:(70)F4da1,a2,a1,a2,a2;a1,a2;x,y,z=1Γa1Γa22∫0∞∫0∞∫0∞e−s+t+vsa1−1ta2−1va2−1×1F1a2;a2;stvx1F1a1;a1;sy1F1a2;a2;vzdsdtdv,Rea1>0,Rea2>0.If we let x=0 in (70), we have(71)F4da1,a2,a1,a2,a2;a1,a2;0,y,z=1Γa1∫0∞e−1−yssa1−1ds×1Γa2∫0∞e−1−zvva2−1dv=11−ya11−za2.Example 4.
If we puty=z in (70), we have(72)F4da1,a2,a1,a2,a2;a1,a2;0,y,y=11−ya1+a2,
which, fora1=a2=1 and y=1/2, we obtain(73)F4d1,1,1,1,1;1,1;0,12,12=4.Example 5.
If we leta1=a2=1 and y=−z in (70), we have(74)F4d1,1,1,1,1;1,1;0,y,−y=11−y2.
Fory=2, we obtain(75)F4d1,1,1,1,1;1,1;0,2,−2=−1.Example 6.
If we takea2=1/2a1 in (70), we get(76)F4da1,a12,a1,a12,a12;a1,a12;0,y,z=11−y1−za1.
Fora1=2, y=4 and z=1/3, we obtain(77)F4d2,1,2,1,1;2,1;0,4,13=16.


## 5. Conclusion

This article has established several new integral representations of the Euler type and Laplace type for some Gauss hypergeometric functions of three variables. The new integral representations are carried out through ten three-variable hypergeometric functions, namelyF4d,F8c,F10d,F17c,F26b,F26d,F29j,F36e,F41b, and F41e. We conclude that when x = 0, mixed results derived in this work will lead to the corresponding results [22] (Gauss two-variable hypergeometric functions). Similarly, in all derived integrals, if we set y=0 and z=0, we obtained several new integral representations of the Euler type for the classical Gauss hypergeometric function. Several numerical examples were given with graphical representation to describe some of our results FAr.



---

*Source: 1914498-2022-08-02.xml*