# Applications of the Generalized Kummer’s Summation Theorem to Transformation Formulas and Generating Functions

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## Abstract

In this paper, we establish two general transformation formulas for Exton’s quadruple hypergeometric functions K 5 and K 12 by application of the generalized Kummer’s summation theorem. Further, a number of generating functions for Jacobi polynomials are also derived as an applications of our main results.

## Abstract

In this paper, we establish two general transformation formulas for Exton’s quadruple hypergeometric functions K5 and K12 by application of the generalized Kummer’s summation theorem. Further, a number of generating functions for Jacobi polynomials are also derived as an applications of our main results.

## 1 Introduction

In our present investigation, we begin by recalling the following definitions:

The Exton’s quadruple hypergeometric functions K5 and K12 [1]:

$K5(a,a,a,a;b1,b1,b2,b2;c1,c2,c3,c4;x,y,z,t)=∑p,q,r,s=0∞(a)p+q+r+s(b1)p+q(b2)r+s xp yq zr ts(c1)p(c2)q(c3)r(c4)s p!q!r!s!$

and

$K12(a,a,a,a;b1,b2,b3,b4;c1,c1,c2,c2;x,y,z,t)=∑p,q,r,s=0∞(a)p+q+r+s(b1)p(b2)q(b3)r(b4)s xp yq zr ts(c1)p+q(c2)r+s p!q!r!s!,$

where (a)n denotes the Pochhammer’s symbol defined by

$(a)n = 1,if n=0a(a+1)(a+2)…(a+n−1),if n=1,2,3,…$

The Exton’s triple hypergeometric functions X4 and X7 [2]:

$X4(a,b;c1,c2,c3;x,y,z)=∑m,n,p=0∞(a)2m+n+p(b)n+p xm yn zp(c1)m(c2)n(c3)p m!n!p!$

and

$X7(a,b1,b2;c1,c2;x,y,z)=∑m,n,p=0∞(a)2m+n+p(b1)n(b2)p xm yn zp(c1)n+p(c2)m m!n!p!.$

The Saran’s triple hypergeometric functions FE and FG [6]:

$FE(a1,a1,a1,b1,b2,b2;c1,c2,c3;x,y,z)=∑m,n,p=0∞(a1)m+n+p(b1)m(b2)n+p xm yn zp(c1)m(c2)n(c3)p m!n!p!$

and

$FG(a1,a1,a1,b1,b2,b3;c1,c2,c2;x,y,z)=∑m,n,p=0∞(a1)m+n+p(b1)m(b2)n(b3)p xm yn zp(c1)m(c2)n+p m!n!p!.$

The Exton’s double hypergeometric function [3]

$XA:B;B′C:D;D′(a) : (b) ; (b′) ;(c) : (d) ; (d′) ;x , y= ∑m,n=0∞((a))2m+n((b))m((b′))n xm yn((c))2m+n((d))m((d′))n m! n!,$

where the symbol ((a))m denotes the product $∏j=1A(aj)m.$

The Jacobi polynomials $Pn(α,β)(x)$[5]

$Pn(α,β)(x)=(1+α)nn! 2F1−n,1+α+β+n;1+α;1−x2.$

In order to obtain our main results, we require the following generalization of the classical Kummer’s summation theorem for the series 2F1(-1) due to Lavoie et al [4]

$2F1 a , b;1+a−b+i;−1 = Γ(12)Γ(1+a−b+i)Γ(1−b)2aΓ(1−b+12(i+|i|))×{AiΓ(12a+12i+12−[1+i2])Γ(1+12a−b+12i)+BiΓ(12a+12i−[i2])Γ(12+12a−b+12i)}$

for (i = 0, ±1, ±2, ±3, ±4, ±5).

where [x] denotes the greatest integer less than or equal to x and |x| denotes the usual absolute value of x. The coefficients Ai and Bi are given respectively in [4]. When i = 0, (10) reduces immediately to the classical Kummer’s theorem [5]

$2F1 a , b;1+a−b;−1 = Γ(1+a−b)Γ(12)2aΓ(1+12a−b)Γ(12a+12).$

We also require the following identities [8]:

$(α)m+n=(α)m(α+m)n$
$∑m=0∞∑n=0∞A(n,m)=∑m=0∞∑n=0mA(n,m−n)$
$(α)m−n=(−1)n(α)m(1−α−m)n, 0≤n≤m$
$(m−n)!=(−1)n m!(−m)n, 0≤n≤m.$

## 2 Main Results

Theorem 1. The following general transformation formulas for K5 holds true.

$K5(a,a,a,a;b′,b′,b,b;c′,c,d,d+i;x,y,z,−z)=∑m=0∞∑n=0∞∑p=0∞(a)m+n+2p(b′)m+n(b)2p xm yn z2p(c′)m(c)n(d)2p m! n! (2p)!×{Ai′22pΓ(12)Γ(d+i)Γ(d+2p)Γ(d+2p+12(i+|i|))Γ(−p+12i+12−[1+i2])Γ(p+d+12i)+Bi′22pΓ(12)Γ(d+i)Γ(d+2p)Γ(d+2p+12(i+|i|))Γ(−p+12i−[i2])Γ(p+d−12+12i)}+∑m=0∞∑n=0∞∑p=0∞(a)m+n+2p+1(b′)m+n(b)2p+1 xm yn z2p+1(c′)m(c)n(d)2p+1 m! n! (2p+1)!×{Ai″22p+1Γ(12)Γ(d+i)Γ(d+2p+1)Γ(d+2p+1+12(i+|i|))Γ(−p+12i−[1+i2])Γ(p+12+d+12i)+Bi″22p+1Γ(12)Γ(d+i)Γ(d+2p+1)Γ(d+2p+1+12(i+|i|))Γ(−p−12+12i−[i2])Γ(p+d+12i)}$

for (i = 0, ±1, ±2, ±3, ±4, ±5).

The coefficients $Ai′$and $Bi′$can be obtained from the tables of Ai and Bi given in [4] by taking a = -2p, b = 1 -d -2 p and the coefficients $Ai″$and $Bi″$can be also obtained from the same tables by taking a = -2p-1, b = -d-2p.

Theorem 2. The following general transformation formulas for K12 holds true.

$K12(a,a,a,a;b′,b,c−i,c;d,d,e,e;x,y,z,−z)=∑m=0∞∑n=0∞∑p=0∞(a)m+n+2p(b′)m(b)n(c−i)2p xm yn z2p(d)m+n(e)2p m! n! (2p)!×{Ei′22pΓ(12)Γ(1+2p−c+i)Γ(1−c)Γ(1−c+12(i+|i|))Γ(−p+12i+12−[1+i2])Γ(1−p−c+12i)+Fi′22pΓ(12)Γ(1−2p−c+i)Γ(1−c)Γ(1−c+12(i+|i|))Γ(−p+12i−[i2])Γ(−p+12−c+12i)}+∑m=0∞∑n=0∞∑p=0∞(a)m+n+2p+1(b′)m(b)n(c−i)2p+1 xm yn z2p+1(d)m+n(e)2p+1 m! n! (2p+1)!×{Ei″22p+1Γ(12)Γ(−2p−c+i)Γ(1−c)Γ(1−c+12(i+|i|))Γ(−p+12i−[1+i2])Γ(−p+12−c+12i)+Fi″22p+1Γ(12)Γ(−2p−c+i)Γ(1−c)Γ(1−c+12(i+|i|))Γ(−p−12+12i−[i2])Γ(−p−c+12i)}$

for (i = 0, ±1, ±2, ±3, ±4, ±5).

The coefficients $Ei′$and $Fi′$can be obtained from the tables of Ai and Bi given in [4] by taking a = - 2p, b = c. Also the coefficients $Ei″$and $Fi″$can be obtained from the same tables by taking a = -2p-1, b = c.

Proof. Denoting the left hand side of (16) by S, expanding K5 in a power series and using the results (12) – (15), then after simplification, we obtain

$S=∑m=0∞∑n=0∞∑p=0∞(a)m+n+p(b′)m+n(b)p xm yn zp(c′)m(c)n(d)p m! n! p!×2F1−p , 1−d−p;d+i;−1$

Separating into even and odd powers of z, we have

$S=∑m=0∞∑n=0∞∑p=0∞(a)m+n+2p(b′)m+n(b)2p xm yn z2p(c′)m(c)n(d)2p m! n! (2p)!2F1−2p , 1−d−2p;d+i;−1+∑m=0∞∑n=0∞∑p=0∞(a)m+n+2p+1(b′)m+n(b)2p+1 xm yn z2p+1(c′)m(c)n(d)2p+1 m! n! (2p+1)!2F1−2p−1 , −d−2p;d+i;−1$

Now,by applying the generalized Kummer’s theorem (10) to each 2F1[-1], then after simplification, we arrive at the right hand side of (16). This completes the proof of (16). The proof of (17) is similar to that of (16) and we use here the result (2).

Remark 1. On taking x = 0 in (16) and (17), we obtain the following transformation formulas for Saran’s triple hypergeometric functions FE and FG:

Corollary 3.

$FE(a,a,a;b′,b,b;c,d,d+i;y,z,−z)=∑n=0∞∑p=0∞(a)n+2p(b′)n(b)2p yn z2p(c)n(d)2p n! (2p)!×{Ai′22pΓ(12)Γ(d+i)Γ(d+2p)Γ(d+2p+12(i+|i|))Γ(−p+12i+12−[1+i2])Γ(p+d+12i)+Bi′22pΓ(12)Γ(d+i)Γ(d+2p)Γ(d+2p+12(i+|i|))Γ(−p+12i−[i2])Γ(p+d−12+12i)}+∑n=0∞∑p=0∞(a)n+2p+1(b′)n(b)2p+1 yn z2p+1(c)n(d)2p+1 n! (2p+1)!×{Ai″22p+1Γ(12)Γ(d+i)Γ(d+2p+1)Γ(d+2p+1+12(i+|i|))Γ(−p+12i−[1+i2])Γ(p+12+d+12i)+Bi″22p+1Γ(12)Γ(d+i)Γ(d+2p+1)Γ(d+2p+1+12(i+|i|))Γ(−p−12+12i−[i2])Γ(p+d+12i)}$

for (i = 0, ±1, ±2, ±3, ±4, ±5).

Corollary 4.

$FG(a,a,a;b,c−i,c;d,e,e;y,z,−z)=∑n=0∞∑p=0∞(a)n+2p(b)n(c−i)2p yn z2p(d)n(e)2p n! (2p)!×{Ei′22pΓ(12)Γ(1−2p−c+i)Γ(1−c)Γ(1−c+12(i+|i|))Γ(−p+12i+12−[1+i2])Γ(1−p−c+12i)+Fi′22pΓ(12)Γ(1−2p−c+i)Γ(1−c)Γ(1−c+12(i+|i|))Γ(−p+12i−[i2])Γ(−p+12−c+12i)}+∑n=0∞∑p=0∞(a)n+2p+1(b)n(c−i)2p+1 yn z2p+1(d)n(e)2p+1 n! (2p+1)!×{Ei″22p+1Γ(12)Γ(−2p−c+i)Γ(1−c)Γ(1−c+12(i+|i|))Γ(−p+12i−[1+i2])Γ(−p+12−c+12i)+Fi″22p+1Γ(12)Γ(−2p−c+i)Γ(1−c)Γ(1−c+12(i+|i|))Γ(−p−12+12i−[i2])Γ(−p−c+12i)}$

for (i = 0, ±1, ±2, ±3, ±4, ±5).

Special cases of (16) and (17)

Here we mention some special cases of our results (16) and (17) and we will use in each case the following results [8]:

$Γ(α+n)Γ(α)=(α)n, Γ(α−n)Γ(α) = (−1)n(1−α)n$
$Γ12Γ(1+α) = 2α Γ12+12αΓ1+12α$
$(α)2n=22n12αn 12α+12n$
$(2n)! = 22n12nn! and (2n+1)! = 22n32nn!.$
• Taking i = 0 and d = b in (16), we get

$K5(a,a,a,a;b′,b′,b,b;c′,c,b,b;x,y,z,−z)=X4(a,b′;b,c′,c;−z2,x,y).$
• Taking i = 1 and d = b-1 in (16), we get

$K5(a,a,a,a;b′,b′,b,b;c′,c,b−1,b;x,y,z,−z)=X4(a,b′;b−1,c′,c;−z2,x,y)+azb−1X4(a+1,b′;b,c′,c;−z2,x,y).$
• Taking i = -1 and d = b in (16), we get

$K5(a,a,a,a;b′,b′,b,b;c′,c,b,b−1;x,y,z,−z)=X4(a,b′;b−1,c′,c;−z2,x,y)−az2X4(a+1,b′;b,c′,c;−z2,x,y).$
• Taking i = 0 and e = 2c in (17), we get

$K12(a,a,a,a;b′,b,c,c;d,d,2c,2c;x,y,z,−z)=X7(a,b′,b;d,c+12;z2/4,x,y).$
• Taking i = 1 and e = 2c-1 in (17), we get

$K12(a,a,a,a;b′,b,c−1,c;d,d,2c−1,2c−1;x,y,z,−z)=X7(a,b′,b;d,c−12;z2/4,x,y)−az2c−1X7(a+1,b′,b;d,c+12;z2/4,x,y).$
• Taking i = -1 and e = 2c+1 in (17), we get

$K12(a,a,a,a;b′,b,c+1,c;d,d,2c+1,2c+1;x,y,z,−z)=X7(a,b′,b;d,c+12;z2/4,x,y)−az2c+1X7(a+1,b′,b;d,c+32;z2/4,x,y).$

## 3 Applications to Generating Functions

Two interesting generating functions for Jacobi polynomials $Pn(α,β)(x)$are given by Sharma and Mittal [7]

$∑n=0∞(λ)n(−α−β)nPn(α−n,β−n)(x)F4(λ+n,γ;δ,ρ;y,z)tn=[1−(1−x)t2]−λ×FE(λ,λ,λ,−α,γ,γ;−α−β,δ,ρ;−2t2−(1−x)t,2y2−(1−x)t,2z2−(1−x)t)$

and

$∑n=0∞(λ)n(−α−β)nPn(α−n,β−n)(x)F1(λ+n,γ,ρ;δ;y,z)tn=[1−(1−x)t2]−λ×FG(λ,λ,λ,−α,γ,ρ;−α−β,δ,δ;−2t2−(1−x)t,2y2−(1−x)t,2z2−(1−x)t),$

where F1 and F4 are Appell’s double hypergeometric functions [8].

Now, in (30), replacing ρ by δ +i and z by -y and using (18), we get the following families of generating functions for Jacobi polynomials:

$∑n=0∞(λ)n(−α−β)nPn(α−n,β−n)(x)F4(λ+n,γ;δ,δ+i;y,−y)tn=[1−(1−x)t2]−λ∑n=0∞∑p=0∞(λ)n+2p(−α)n(γ)2p(−α−β)n(δ)2p n! (2p)![−2t2−(1−x)t]n[2y2−(1−x)t]2p×{Ai′22pΓ(12)Γ(δ+i)Γ(δ+2p)Γ(δ+2p+12(i+|i|))Γ(−p+12i+12−[1+i2])Γ(p+δ+12i)+Bi′22pΓ(12)Γ(δ+i)Γ(δ+2p)Γ(δ+2p+12(i+|i|))Γ(−p+12i−[i2])Γ(p+δ−12+12i)}+∑n=0∞∑p=0∞(λ)n+2p+1(−α)n(γ)2p+1(−α−β)n(δ)2p+1 n! (2p+1)![−2t2−(1−x)t]n[2y2−(1−x)t]2p+1×{Ai″22p+1Γ(12)Γ(δ+i)Γ(δ+2p+1)Γ(δ+2p+1+12(i+|i|))Γ(−p+12i−[1+i2])Γ(p+12+δ+12i)+Bi″22p+1Γ(12)Γ(δ+i)Γ(δ+2p+1)Γ(δ+2p+1+12(i+|i|))Γ(−p−12+12i−[i2])Γ(p+δ+12i)}$

for (i = 0, ±1, ±2, ±3, ±4, ±5).

Next, in (31), replacing γ by ρ -i and z by -y and using (19), we get the following families of generating functions for Jacobi polynomials:

$∑n=0∞(λ)n(−α−β)nPn(α−n,β−n)(x)F1(λ+n,ρ−i,ρ;δ;y,−y)tn=[1−(1−x)t2]−λ∑n=0∞∑p=0∞(λ)n+2p(−α)n(ρ−i)2p(−α−β)n(δ)2p n! (2p)![−2t2−(1−x)t]n[2y2−(1−x)t]2p×{Ei′22pΓ(12)Γ(1−2p−ρ+i)Γ(1−ρ)Γ(1−ρ+12(i+|i|))Γ(−p+12i+12−[1+i2])Γ(1−p−ρ+12i)+Fi′22pΓ(12)Γ(1−2p−ρ+i)Γ(1−ρ)Γ(1−ρ+12(i+|i|))Γ(−p+12i−[i2])Γ(−p+12−ρ+12i)}+∑n=0∞∑p=0∞(λ)n+2p+1(−α)n(ρ−i)2p+1(−α−β)n(δ)2p+1 n! (2p+1)![−2t2−(1−x)t]n[2y2−(1−x)t]2p+1×{Ei″22p+1Γ(12)Γ(−2p−ρ+i)Γ(1−ρ)Γ(1−ρ+12(i+|i|))Γ(−p+12i−[1+i2])Γ(−p+12−ρ+12i)+Fi″22p+1Γ(12)Γ(−2p−ρ+i)Γ(1−ρ)Γ(1−ρ+12(i+|i|))Γ(−p−12+12i−[i2])Γ(−p−ρ+12i)}$

for (i = 0, ±1, ±2, ±3, ±4, ±5).

Now, we mention some interesting special cases of the results (32) and (33) and we using in each case the results (20)–(23).

• Taking i = 0 in (32) and (33), we get

$∑n=0∞(λ)n(−α−β)nPn(α−n,β−n)(x)F4(λ+n,γ;δ,δ;y,−y)tn=(1−(1−x)t2)−λX1:2;10:3;1λ:12γ,12γ+12;−α;−:δ,12δ,12δ+12;−α−β;−(2y2−(1−x)t)2,−2t2−(1−x)t$

and

$∑n=0∞(λ)n(−α−β)nPn(α−n,β−n)(x)F1(λ+n,ρ,ρ;δ;y,−y)tn=(1−(1−x)t2)−λX1:1;10:2;1λ:ρ;−α;−:12δ,12δ+12;−α−β;(y2−(1−x)t)2,−2t2−(1−x)t.$
• Taking i = 1 in (32) and (33), we get

$∑n=0∞(λ)n(−α−β)nPn(α−n,β−n)(x)F4(λ+n,γ;δ,δ+1;y,−y)tn=(1−(1−x)t2)−λ{X1:2;10:3;1λ:12γ,12γ+12;−α;−:δ,12δ+12,12δ+1;−α−β;−(2y2−(1−x)t)2,−2t2−(1−x)t+2λγyδ(δ+1)(2−t+xt)×X1:2;10:3;1λ+1:12γ+12,12γ+1;−α;−:δ+1,12δ+1,12δ+32;−α−β;−(2y2−(1−x)t)2,−2t2−(1−x)t}$

and

$∑n=0∞(λ)n(−α−β)nPn(α−n,β−n)(x)F1(λ+n,ρ−1,ρ;δ;y,−y)tn=(1−(1−x)t2)−λ{X1:1;10:2;1λ:ρ;−α;−:12δ,12δ+12;−α−β;(y2−(1−x)t)2,−2t2−(1−x)t+2λyδ(2−t+xt)X1:1;10:2;1λ+1:ρ;−α;−:12δ+12,12δ+1;−α−β;(y2−(1−x)t)2,−2t2−(1−x)t}.$
• Taking i = -1 in (32) and (33), we get

$∑n=0∞(λ)n(−α−β)nPn(α−n,β−n)(x)F4(λ+n,γ;δ,δ−1;y,−y)tn=(1−(1−x)t2)−λ{X1:2;10:3;1λ:12γ,12γ+12;−α;−:δ−1,12δ,12δ+12;−α−β;−(2y2−(1−x)t)2,−2t2−(1−x)t−λγyδ(2−t+xt)X1:2;10:3;1λ+1:12γ+12,12γ+1;−α;−:δ,12δ+12,δ+1;−α−β;−(2y2−(1−x)t)2,−2t2−(1−x)t}$

and

$∑n=0∞(λ)n(−α−β)nPn(α−n,β−n)(x)F1(λ+n,ρ+1,ρ;δ;y,−y)tn=(1−(1−x)t2)−λ{X1:1;10:2;1λ:ρ+1;−α;−:12δ,12δ+12;−α−β;(y2−(1−x)t)2,−2t2−(1−x)t+2λyδ(2−t+xt)X1:1;10:2;1λ+1:ρ+1;−α;−:12δ+12,12δ+1;−α−β;(y2−(1−x)t)2,−2t2−(1−x)t}.$

The other special cases of (32) and (33) can also be obtained in the similar manner.

Communicated by Juan L.G. Guirao

## References

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H. Exton, (1976), Multiple Hypergeometric Functions and Applications, John Wiley and Sons, Halsted Press, New York.

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H. Exton, (1982), Hypergeometric functions of three variables, J. Indian Acad. Maths. 4, 113–119.

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[1]

H. Exton, (1976), Multiple Hypergeometric Functions and Applications, John Wiley and Sons, Halsted Press, New York.

[2]

H. Exton, (1982), Hypergeometric functions of three variables, J. Indian Acad. Maths. 4, 113–119.

[3]

H. Exton, (1982), Reducible double hypergeometric functions and associated integrals, An Fac. Ci. Univ. Porto, 63, 137–143.

[4]

J. L. Lavoie, F. Grondin and A. K. Rathie, (1996), Generalizations of Whipple’s theorem on the sum of a 3F2 Journal of Computational and Applied Mathematics, 72, 293–300. 10.1016/0377-0427(95)00279-0.

[5]

E. D. Rainville, (1960), Special Functions, The Macmillan Company, New York.

[6]

S. Saran, (1954), Hypergeometric functions of three variables, Ganita 5, 77–91.

[7]

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# Applied Mathematics and Nonlinear Sciences

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