1 Introduction
In our present investigation, we begin by recalling the following definitions:
The Exton’s quadruple hypergeometric functions K 5 and K 12 [1 ]:
(1) K 5 ( a , a , a , a ; b 1 , b 1 , b 2 , b 2 ; c 1 , c 2 , c 3 , c 4 ; x , y , z , t ) = ∑ p , q , r , s = 0 ∞ ( a ) p + q + r + s ( b 1 ) p + q ( b 2 ) r + s x p y q z r t s ( c 1 ) p ( c 2 ) q ( c 3 ) r ( c 4 ) s p ! q ! r ! s ! $$\begin{equation}\label{equation1} K_5(a,a,a,a;b_1,b_1,b_2,b_2;c_1,c_2,c_3,c_4;x,y,z,t) =\sum_{p,q,r,s=0}^\infty\frac{(a)_{p+q+r+s}(b_1)_{p+q}(b_2)_{r+s}~x^p~ y^q~ z^r~ t^s}{(c_1)_p (c_2)_q (c_3)_r (c_4)_s~ p! q! r! s!} \end{equation}$$ and
(2) K 12 ( a , a , a , a ; b 1 , b 2 , b 3 , b 4 ; c 1 , c 1 , c 2 , c 2 ; x , y , z , t ) = ∑ p , q , r , s = 0 ∞ ( a ) p + q + r + s ( b 1 ) p ( b 2 ) q ( b 3 ) r ( b 4 ) s x p y q z r t s ( c 1 ) p + q ( c 2 ) r + s p ! q ! r ! s ! , $$\begin{equation} \label{equation2} K_{12}(a,a,a,a;b_1,b_2,b_3,b_4;c_1,c_1,c_2,c_2;x,y,z,t) =\sum_{p,q,r,s=0}^\infty\frac{(a)_{p+q+r+s}(b_1)_p(b_2)_q (b_3)_r (b_4)_s~x^p~ y^q~ z^r~ t^s}{(c_1)_{p+q} (c_2)_{r+s}~ p! q! r! s!}, \end{equation}$$ where (a )n denotes the Pochhammer’s symbol defined by
(3) ( a ) n = 1 , i f n = 0 a ( a + 1 ) ( a + 2 ) … ( a + n − 1 ) , i f n = 1 , 2 , 3 , … $$\begin{equation}\label{equation3} (a)_n~=~\left \{ \begin{array}{cl} 1&, if~ n=0\\ a(a+1)(a+2)\dots(a+n-1)&,if~ n = 1,2,3,\dots\end{array} \right. \end{equation}$$ The Exton’s triple hypergeometric functions X 4 and X 7 [2 ]:
(4) X 4 ( a , b ; c 1 , c 2 , c 3 ; x , y , z ) = ∑ m , n , p = 0 ∞ ( a ) 2 m + n + p ( b ) n + p x m y n z p ( c 1 ) m ( c 2 ) n ( c 3 ) p m ! n ! p ! $$\begin{equation}\label{equation4} X_4(a,b;c_1,c_2,c_3;x,y,z) =\sum_{m,n,p=0}^\infty\frac{(a)_{2m+n+p}(b)_{n+p}~x^m~ y^n~ z^p}{(c_1)_m (c_2)_n(c_3)_p~ m! n! p!} \end{equation}$$ and
(5) X 7 ( a , b 1 , b 2 ; c 1 , c 2 ; x , y , z ) = ∑ m , n , p = 0 ∞ ( a ) 2 m + n + p ( b 1 ) n ( b 2 ) p x m y n z p ( c 1 ) n + p ( c 2 ) m m ! n ! p ! . $$\begin{equation}\label{equation5} X_7(a,b_1,b_2;c_1,c_2;x,y,z) =\sum_{m,n,p=0}^\infty\frac{(a)_{2m+n+p}(b_1)_n(b_2)_p~x^m~ y^n~ z^p}{(c_1)_{n+p} (c_2)_m~ m! n! p!}. \end{equation}$$ The Saran’s triple hypergeometric functions FE and FG [6 ]:
(6) F E ( a 1 , a 1 , a 1 , b 1 , b 2 , b 2 ; c 1 , c 2 , c 3 ; x , y , z ) = ∑ m , n , p = 0 ∞ ( a 1 ) m + n + p ( b 1 ) m ( b 2 ) n + p x m y n z p ( c 1 ) m ( c 2 ) n ( c 3 ) p m ! n ! p ! $$\begin{equation}\label{equation6} F_E(a_1,a_1,a_1,b_1,b_2,b_2;c_1,c_2,c_3;x,y,z)=\sum_{m,n,p=0}^\infty\frac{(a_1)_{m+n+p}(b_1)_m(b_2)_{n+p}~x^m~ y^n~ z^p}{(c_1)_m (c_2)_n(c_3)_p~ m! n! p!} \end{equation}$$ and
(7) F G ( a 1 , a 1 , a 1 , b 1 , b 2 , b 3 ; c 1 , c 2 , c 2 ; x , y , z ) = ∑ m , n , p = 0 ∞ ( a 1 ) m + n + p ( b 1 ) m ( b 2 ) n ( b 3 ) p x m y n z p ( c 1 ) m ( c 2 ) n + p m ! n ! p ! . $$\begin{equation}\label{equation7} F_G(a_1,a_1,a_1,b_1,b_2,b_3;c_1,c_2,c_2;x,y,z)=\sum_{m,n,p=0}^\infty\frac{(a_1)_{m+n+p}(b_1)_m(b_2)_n(b_3)_p~x^m~ y^n~ z^p}{(c_1)_m (c_2)_{n+p}~ m! n! p!}. \end{equation}$$ The Exton’s double hypergeometric function [3 ]
(8) X A : B ; B ′ C : D ; D ′ ( a ) : ( b ) ; ( b ′ ) ; ( c ) : ( d ) ; ( d ′ ) ; x , y = ∑ m , n = 0 ∞ ( ( a ) ) 2 m + n ( ( b ) ) m ( ( b ′ ) ) n x m y n ( ( c ) ) 2 m + n ( ( d ) ) m ( ( d ′ ) ) n m ! n ! , $$\begin{equation}\label{equation8} X \begin{array}{c} A:B;B'\\ C:D;D' \end{array} \left [ \begin{array}{c} (a)~:~(b)~;~(b')~;\\ (c)~:~(d)~;~(d')~; \end{array} \begin{array}{c} x~,~y \end{array} \right] =~ \sum_{m,n=0}^{\infty}\frac{((a))_{2m+n}((b))_m((b'))_n ~x^m~y^n}{((c))_{2m+n}((d))_m((d'))_n ~m!~n!}, \end{equation}$$ where the symbol ((a ))m ∏ j = 1 A ( a j ) m . $ \prod\limits_{j=1}^{A}(a_j)_m $.
The Jacobi polynomials P n ( α , β ) ( x ) $P_n^{(\alpha,\beta)}(x)$ 5 ]
(9) P n ( α , β ) ( x ) = ( 1 + α ) n n ! 2 F 1 − n , 1 + α + β + n ; 1 + α ; 1 − x 2 . $$\begin{equation}P_n^{(\alpha,\beta)}(x) = \frac{(1+\alpha)_n}{n!}~{}_2F_1 \left [ \begin{array}{cc} -n,1+\alpha+\beta+n&;\\ 1+\alpha&; \end{array} \begin{array}{c} \frac{1-x}{2} \end{array} \right]. \end{equation}$$ In order to obtain our main results, we require the following generalization of the classical Kummer’s summation theorem for the series 2 F 1 (- 1) due to Lavoie et al [4 ]
(10) 2 F 1 a , b ; 1 + a − b + i ; − 1 = Γ ( 1 2 ) Γ ( 1 + a − b + i ) Γ ( 1 − b ) 2 a Γ ( 1 − b + 1 2 ( i + | i | ) ) × { A i Γ ( 1 2 a + 1 2 i + 1 2 − [ 1 + i 2 ] ) Γ ( 1 + 1 2 a − b + 1 2 i ) + B i Γ ( 1 2 a + 1 2 i − [ i 2 ] ) Γ ( 1 2 + 1 2 a − b + 1 2 i ) } $$\begin{equation} {}_2F_1~ \left [ \begin{array}{cc} a~,~b&;\\ 1+a-b+i&; \end{array} \begin{array}{c} -1 \end{array} \right]~=~ \frac{\Gamma(\frac{1}{2})\Gamma(1+a-b+i)\Gamma(1-b)}{2^a\Gamma(1-b+\frac{1}{2}(i+|i|))} \nonumber \\ \times \biggr \{ \frac{A_i}{\Gamma(\frac{1}{2}a+\frac{1}{2}i+\frac{1}{2}-[\frac{1+i}{2}])\Gamma(1+\frac{1}{2}a-b+\frac{1}{2}i)} +\frac{B_i}{\Gamma(\frac{1}{2}a+\frac{1}{2}i-[\frac{i}{2}])\Gamma(\frac{1}{2}+\frac{1}{2}a-b+\frac{1}{2}i)} \biggl\} \end{equation}$$ 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 ]
(11) 2 F 1 a , b ; 1 + a − b ; − 1 = Γ ( 1 + a − b ) Γ ( 1 2 ) 2 a Γ ( 1 + 1 2 a − b ) Γ ( 1 2 a + 1 2 ) . $$\begin{equation}\label{equation11} {}_2F_1~ \left [ \begin{array}{cc} a~,~b&;\\ 1+a-b&; \end{array} \begin{array}{c} -1 \end{array} \right]~=~ \frac{\Gamma(1+a-b)\Gamma(\frac{1}{2})}{2^a\Gamma(1+\frac{1}{2}a-b)\Gamma(\frac{1}{2}a+\frac{1}{2})}. \end{equation}$$ We also require the following identities [8 ]:
(12) ( α ) m + n = ( α ) m ( α + m ) n $$\begin{equation}\label{equation12} (\alpha)_{m+n} = (\alpha)_m(\alpha+m)_n \end{equation}$$ (13) ∑ m = 0 ∞ ∑ n = 0 ∞ A ( n , m ) = ∑ m = 0 ∞ ∑ n = 0 m A ( n , m − n ) $$\begin{equation}\label{equation13} \sum_{m=0}^{\infty}\sum_{n=0}^{\infty}A(n,m)=\sum_{m=0}^{\infty}\sum_{n=0}^{m}A(n,m-n) \end{equation}$$ (14) ( α ) m − n = ( − 1 ) n ( α ) m ( 1 − α − m ) n , 0 ≤ n ≤ m $$\begin{equation}\label{equation14} (\alpha)_{m-n} = \frac{(-1)^n(\alpha)_m}{(1-\alpha-m)_n},~~~0 \leq n \leq m \end{equation}$$ (15) ( m − n ) ! = ( − 1 ) n m ! ( − m ) n , 0 ≤ n ≤ m . $$\begin{equation}\label{equation15} (m-n)! = \frac{(-1)^n~m!}{(-m)_n},~~~0 \leq n \leq m. \end{equation}$$ 2 Main Results
Theorem 1. The following general transformation formulas for K 5 holds true .
(16) K 5 ( 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 + 2 p ( b ′ ) m + n ( b ) 2 p x m y n z 2 p ( c ′ ) m ( c ) n ( d ) 2 p m ! n ! ( 2 p ) ! × { A i ′ 2 2 p Γ ( 1 2 ) Γ ( d + i ) Γ ( d + 2 p ) Γ ( d + 2 p + 1 2 ( i + | i | ) ) Γ ( − p + 1 2 i + 1 2 − [ 1 + i 2 ] ) Γ ( p + d + 1 2 i ) + B i ′ 2 2 p Γ ( 1 2 ) Γ ( d + i ) Γ ( d + 2 p ) Γ ( d + 2 p + 1 2 ( i + | i | ) ) Γ ( − p + 1 2 i − [ i 2 ] ) Γ ( p + d − 1 2 + 1 2 i ) } + ∑ m = 0 ∞ ∑ n = 0 ∞ ∑ p = 0 ∞ ( a ) m + n + 2 p + 1 ( b ′ ) m + n ( b ) 2 p + 1 x m y n z 2 p + 1 ( c ′ ) m ( c ) n ( d ) 2 p + 1 m ! n ! ( 2 p + 1 ) ! × { A i ″ 2 2 p + 1 Γ ( 1 2 ) Γ ( d + i ) Γ ( d + 2 p + 1 ) Γ ( d + 2 p + 1 + 1 2 ( i + | i | ) ) Γ ( − p + 1 2 i − [ 1 + i 2 ] ) Γ ( p + 1 2 + d + 1 2 i ) + B i ″ 2 2 p + 1 Γ ( 1 2 ) Γ ( d + i ) Γ ( d + 2 p + 1 ) Γ ( d + 2 p + 1 + 1 2 ( i + | i | ) ) Γ ( − p − 1 2 + 1 2 i − [ i 2 ] ) Γ ( p + d + 1 2 i ) } $$\begin{equation} K_5(a,a,a,a;b',b',b,b;c',c,d,d+i;x,y,z,-z) \nonumber \\ =\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\sum_{p=0}^{\infty}\frac{(a)_{m+n+2p}(b')_{m+n}(b)_{2p}~x^m~y^n~z^{2p}}{(c')_m(c)_n(d)_{2p}~m!~n!~(2p)!} \nonumber \\ \times \biggr \{ A'_i \frac{2^{2p}\Gamma(\frac{1}{2})\Gamma(d+i)\Gamma(d+2p)}{\Gamma(d+2p+\frac{1}{2}(i+|i|))\Gamma(-p+\frac{1}{2}i+\frac{1}{2}-\big [\frac{1+i}{2}\big ])\Gamma(p+d+\frac{1}{2}i)} \nonumber \\ +B'_i\frac{2^{2p}\Gamma(\frac{1}{2})\Gamma(d+i)\Gamma(d+2p)}{\Gamma(d+2p+\frac{1}{2}(i+|i|))\Gamma(-p+\frac{1}{2}i-\big [\frac{i}{2}\big ])\Gamma(p+d-\frac{1}{2}+\frac{1}{2}i)}\biggl \} \nonumber \\ +\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\sum_{p=0}^{\infty}\frac{(a)_{m+n+2p+1}(b')_{m+n}(b)_{2p+1}~x^m~y^n~z^{2p+1}}{(c')_m(c)_n(d)_{2p+1}~m!~n!~(2p+1)!}\nonumber \\ \times \biggr \{ A''_i \frac{2^{2p+1}\Gamma(\frac{1}{2})\Gamma(d+i)\Gamma(d+2p+1)}{\Gamma(d+2p+1+\frac{1}{2}(i+|i|))\Gamma(-p+\frac{1}{2}i-\big [\frac{1+i}{2}\big ])\Gamma(p+\frac{1}{2}+d+\frac{1}{2}i)} \nonumber \\ +B''_i\frac{2^{2p+1}\Gamma(\frac{1}{2})\Gamma(d+i)\Gamma(d+2p+1)}{\Gamma(d+2p+1+\frac{1}{2}(i+|i|))\Gamma(-p-\frac{1}{2}+\frac{1}{2}i-\big [\frac{i}{2}\big ])\Gamma(p+d+\frac{1}{2}i)} \biggl \} \end{equation}$$ for (i = 0, ±1, ±2, ±3, ±4, ±5).
The coefficients A i ′ $ A'_i $ and B i ′ $B'_i$ 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 A i ″ $A''_i$ and B i ″ $B''_i$ can be also obtained from the same tables by taking a = - 2p- 1, b = -d- 2p .
Theorem 2. The following general transformation formulas for K 12 holds true .
(17) K 12 ( 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 + 2 p ( b ′ ) m ( b ) n ( c − i ) 2 p x m y n z 2 p ( d ) m + n ( e ) 2 p m ! n ! ( 2 p ) ! × { E i ′ 2 2 p Γ ( 1 2 ) Γ ( 1 + 2 p − c + i ) Γ ( 1 − c ) Γ ( 1 − c + 1 2 ( i + | i | ) ) Γ ( − p + 1 2 i + 1 2 − [ 1 + i 2 ] ) Γ ( 1 − p − c + 1 2 i ) + F i ′ 2 2 p Γ ( 1 2 ) Γ ( 1 − 2 p − c + i ) Γ ( 1 − c ) Γ ( 1 − c + 1 2 ( i + | i | ) ) Γ ( − p + 1 2 i − [ i 2 ] ) Γ ( − p + 1 2 − c + 1 2 i ) } + ∑ m = 0 ∞ ∑ n = 0 ∞ ∑ p = 0 ∞ ( a ) m + n + 2 p + 1 ( b ′ ) m ( b ) n ( c − i ) 2 p + 1 x m y n z 2 p + 1 ( d ) m + n ( e ) 2 p + 1 m ! n ! ( 2 p + 1 ) ! × { E i ″ 2 2 p + 1 Γ ( 1 2 ) Γ ( − 2 p − c + i ) Γ ( 1 − c ) Γ ( 1 − c + 1 2 ( i + | i | ) ) Γ ( − p + 1 2 i − [ 1 + i 2 ] ) Γ ( − p + 1 2 − c + 1 2 i ) + F i ″ 2 2 p + 1 Γ ( 1 2 ) Γ ( − 2 p − c + i ) Γ ( 1 − c ) Γ ( 1 − c + 1 2 ( i + | i | ) ) Γ ( − p − 1 2 + 1 2 i − [ i 2 ] ) Γ ( − p − c + 1 2 i ) } $$\begin{equation} K_{12}(a,a,a,a;b',b,c-i,c;d,d,e,e;x,y,z,-z) \nonumber \\ =\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\sum_{p=0}^{\infty}\frac{(a)_{m+n+2p}(b')_m(b)_n(c-i)_{2p}~x^m~y^n~z^{2p}}{(d)_{m+n}(e)_{2p}~m!~n!~(2p)!} \nonumber \\ \times \biggr \{ E'_i \frac{2^{2p}\Gamma(\frac{1}{2})\Gamma(1+2p-c+i)\Gamma(1-c)}{\Gamma(1-c+\frac{1}{2}(i+|i|))\Gamma(-p+\frac{1}{2}i+\frac{1}{2}-\big [\frac{1+i}{2}\big ])\Gamma(1-p-c+\frac{1}{2}i)} \nonumber \\ +F'_i\frac{2^{2p}\Gamma(\frac{1}{2})\Gamma(1-2p-c+i)\Gamma(1-c)}{\Gamma(1-c+\frac{1}{2}(i+|i|))\Gamma(-p+\frac{1}{2}i-\big [\frac{i}{2}\big ])\Gamma(-p+\frac{1}{2}-c+\frac{1}{2}i)}\biggl \} \nonumber \\ +\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\sum_{p=0}^{\infty}\frac{(a)_{m+n+2p+1}(b')_m(b)_n(c-i)_{2p+1}~x^m~y^n~z^{2p+1}}{(d)_{m+n}(e)_{2p+1}~m!~n!~(2p+1)!} \nonumber \\ \times \biggr \{ E''_i \frac{2^{2p+1}\Gamma(\frac{1}{2})\Gamma(-2p-c+i)\Gamma(1-c)}{\Gamma(1-c+\frac{1}{2}(i+|i|))\Gamma(-p+\frac{1}{2}i-\big [\frac{1+i}{2}\big ])\Gamma(-p+\frac{1}{2}-c+\frac{1}{2}i)} \nonumber \\ +F''_i\frac{2^{2p+1}\Gamma(\frac{1}{2})\Gamma(-2p-c+i)\Gamma(1-c)}{\Gamma(1-c+\frac{1}{2}(i+|i|))\Gamma(-p-\frac{1}{2}+\frac{1}{2}i-\big [\frac{i}{2}\big ])\Gamma(-p-c+\frac{1}{2}i)} \biggl \} \end{equation} $$ for (i = 0, ±1, ±2, ±3, ±4, ±5).
The coefficients E i ′ $E'_i$ and F i ′ $F'_i$ can be obtained from the tables of Ai and Bi given in [4] by taking a = - 2p, b = c. Also the coefficients E i ″ $E''_i$ and F i ″ $F''_i$ 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 K 5 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 x m y n z p ( c ′ ) m ( c ) n ( d ) p m ! n ! p ! × 2 F 1 − p , 1 − d − p ; d + i ; − 1 $$\begin{equation*} S=\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\sum_{p=0}^{\infty}\frac{(a)_{m+n+p}(b')_{m+n}(b)_p~x^m~y^n~z^p}{(c')_m(c)_n(d)_p~m!~n!~p!} \times {}_2F_1\left [ \begin{array}{cc} -p~,~1-d-p&;\\ d+i&; \end{array} \begin{array}{c} -1 \end{array} \right] \end{equation*}$$ Separating into even and odd powers of z , we have
S = ∑ m = 0 ∞ ∑ n = 0 ∞ ∑ p = 0 ∞ ( a ) m + n + 2 p ( b ′ ) m + n ( b ) 2 p x m y n z 2 p ( c ′ ) m ( c ) n ( d ) 2 p m ! n ! ( 2 p ) ! 2 F 1 − 2 p , 1 − d − 2 p ; d + i ; − 1 + ∑ m = 0 ∞ ∑ n = 0 ∞ ∑ p = 0 ∞ ( a ) m + n + 2 p + 1 ( b ′ ) m + n ( b ) 2 p + 1 x m y n z 2 p + 1 ( c ′ ) m ( c ) n ( d ) 2 p + 1 m ! n ! ( 2 p + 1 ) ! 2 F 1 − 2 p − 1 , − d − 2 p ; d + i ; − 1 $$\begin{equation*} S=\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\sum_{p=0}^{\infty}\frac{(a)_{m+n+2p}(b')_{m+n}(b)_{2p}~x^m~y^n~z^{2p}}{(c')_m(c)_n(d)_{2p}~m!~n!~(2p)!}{}_2F_1\left [ \begin{array}{cc} -2p~,~1-d-2p&;\\ d+i&; \end{array} \begin{array}{c} -1 \end{array} \right] \\ +\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\sum_{p=0}^{\infty}\frac{(a)_{m+n+2p+1}(b')_{m+n}(b)_{2p+1}~x^m~y^n~z^{2p+1}}{(c')_m(c)_n(d)_{2p+1}~m!~n!~(2p+1)!} {}_2F_1\left [ \begin{array}{cc} -2p-1~,~-d-2p&;\\ d+i&; \end{array} \begin{array}{c} -1 \end{array} \right] \end{equation*}$$ Now,by applying the generalized Kummer’s theorem (10) to each 2 F 1 [- 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.
(18) F E ( a , a , a ; b ′ , b , b ; c , d , d + i ; y , z , − z ) = ∑ n = 0 ∞ ∑ p = 0 ∞ ( a ) n + 2 p ( b ′ ) n ( b ) 2 p y n z 2 p ( c ) n ( d ) 2 p n ! ( 2 p ) ! × { A i ′ 2 2 p Γ ( 1 2 ) Γ ( d + i ) Γ ( d + 2 p ) Γ ( d + 2 p + 1 2 ( i + | i | ) ) Γ ( − p + 1 2 i + 1 2 − [ 1 + i 2 ] ) Γ ( p + d + 1 2 i ) + B i ′ 2 2 p Γ ( 1 2 ) Γ ( d + i ) Γ ( d + 2 p ) Γ ( d + 2 p + 1 2 ( i + | i | ) ) Γ ( − p + 1 2 i − [ i 2 ] ) Γ ( p + d − 1 2 + 1 2 i ) } + ∑ n = 0 ∞ ∑ p = 0 ∞ ( a ) n + 2 p + 1 ( b ′ ) n ( b ) 2 p + 1 y n z 2 p + 1 ( c ) n ( d ) 2 p + 1 n ! ( 2 p + 1 ) ! × { A i ″ 2 2 p + 1 Γ ( 1 2 ) Γ ( d + i ) Γ ( d + 2 p + 1 ) Γ ( d + 2 p + 1 + 1 2 ( i + | i | ) ) Γ ( − p + 1 2 i − [ 1 + i 2 ] ) Γ ( p + 1 2 + d + 1 2 i ) + B i ″ 2 2 p + 1 Γ ( 1 2 ) Γ ( d + i ) Γ ( d + 2 p + 1 ) Γ ( d + 2 p + 1 + 1 2 ( i + | i | ) ) Γ ( − p − 1 2 + 1 2 i − [ i 2 ] ) Γ ( p + d + 1 2 i ) } $$\begin{equation} F_E(a,a,a;b',b,b;c,d,d+i;y,z,-z) \nonumber =\sum_{n=0}^{\infty}\sum_{p=0}^{\infty}\frac{(a)_{n+2p}(b')_n(b)_{2p}~~y^n~z^{2p}}{(c)_n(d)_{2p}~n!~(2p)!} \nonumber \\ \times \biggr \{ A'_i \frac{2^{2p}\Gamma(\frac{1}{2})\Gamma(d+i)\Gamma(d+2p)}{\Gamma(d+2p+\frac{1}{2}(i+|i|))\Gamma(-p+\frac{1}{2}i+\frac{1}{2}-\big [\frac{1+i}{2}\big ])\Gamma(p+d+\frac{1}{2}i)} \nonumber \\ +B'_i\frac{2^{2p}\Gamma(\frac{1}{2})\Gamma(d+i)\Gamma(d+2p)}{\Gamma(d+2p+\frac{1}{2}(i+|i|))\Gamma(-p+\frac{1}{2}i-\big [\frac{i}{2}\big ])\Gamma(p+d-\frac{1}{2}+\frac{1}{2}i)}\biggl \} \nonumber \\ +\sum_{n=0}^{\infty}\sum_{p=0}^{\infty}\frac{(a)_{n+2p+1}(b')_n(b)_{2p+1}~y^n~z^{2p+1}}{(c)_n(d)_{2p+1}~n!~(2p+1)!} \nonumber \\ \times \biggr \{ A''_i \frac{2^{2p+1}\Gamma(\frac{1}{2})\Gamma(d+i)\Gamma(d+2p+1)}{\Gamma(d+2p+1+\frac{1}{2}(i+|i|))\Gamma(-p+\frac{1}{2}i-\big [\frac{1+i}{2}\big ])\Gamma(p+\frac{1}{2}+d+\frac{1}{2}i)} \nonumber \\ +B''_i\frac{2^{2p+1}\Gamma(\frac{1}{2})\Gamma(d+i)\Gamma(d+2p+1)}{\Gamma(d+2p+1+\frac{1}{2}(i+|i|))\Gamma(-p-\frac{1}{2}+\frac{1}{2}i-\big [\frac{i}{2}\big ])\Gamma(p+d+\frac{1}{2}i)} \biggl \} \end{equation}$$ for (i = 0, ±1, ±2, ±3, ±4, ±5).
Corollary 4.
(19) F G ( a , a , a ; b , c − i , c ; d , e , e ; y , z , − z ) = ∑ n = 0 ∞ ∑ p = 0 ∞ ( a ) n + 2 p ( b ) n ( c − i ) 2 p y n z 2 p ( d ) n ( e ) 2 p n ! ( 2 p ) ! × { E i ′ 2 2 p Γ ( 1 2 ) Γ ( 1 − 2 p − c + i ) Γ ( 1 − c ) Γ ( 1 − c + 1 2 ( i + | i | ) ) Γ ( − p + 1 2 i + 1 2 − [ 1 + i 2 ] ) Γ ( 1 − p − c + 1 2 i ) + F i ′ 2 2 p Γ ( 1 2 ) Γ ( 1 − 2 p − c + i ) Γ ( 1 − c ) Γ ( 1 − c + 1 2 ( i + | i | ) ) Γ ( − p + 1 2 i − [ i 2 ] ) Γ ( − p + 1 2 − c + 1 2 i ) } + ∑ n = 0 ∞ ∑ p = 0 ∞ ( a ) n + 2 p + 1 ( b ) n ( c − i ) 2 p + 1 y n z 2 p + 1 ( d ) n ( e ) 2 p + 1 n ! ( 2 p + 1 ) ! × { E i ″ 2 2 p + 1 Γ ( 1 2 ) Γ ( − 2 p − c + i ) Γ ( 1 − c ) Γ ( 1 − c + 1 2 ( i + | i | ) ) Γ ( − p + 1 2 i − [ 1 + i 2 ] ) Γ ( − p + 1 2 − c + 1 2 i ) + F i ″ 2 2 p + 1 Γ ( 1 2 ) Γ ( − 2 p − c + i ) Γ ( 1 − c ) Γ ( 1 − c + 1 2 ( i + | i | ) ) Γ ( − p − 1 2 + 1 2 i − [ i 2 ] ) Γ ( − p − c + 1 2 i ) } $$\begin{equation} F_G(a,a,a;b,c-i,c;d,e,e;y,z,-z) \nonumber \\ =\sum_{n=0}^{\infty}\sum_{p=0}^{\infty}\frac{(a)_{n+2p}(b)_n(c-i)_{2p}~~y^n~z^{2p}}{(d)_n(e)_{2p}~n!~(2p)!} \nonumber \\ \times \biggr \{ E'_i \frac{2^{2p}\Gamma(\frac{1}{2})\Gamma(1-2p-c+i)\Gamma(1-c)}{\Gamma(1-c+\frac{1}{2}(i+|i|))\Gamma(-p+\frac{1}{2}i+\frac{1}{2}-\big [\frac{1+i}{2}\big ])\Gamma(1-p-c+\frac{1}{2}i)}\nonumber \\ +F'_i\frac{2^{2p}\Gamma(\frac{1}{2})\Gamma(1-2p-c+i)\Gamma(1-c)}{\Gamma(1-c+\frac{1}{2}(i+|i|))\Gamma(-p+\frac{1}{2}i-\big [\frac{i}{2}\big ])\Gamma(-p+\frac{1}{2}-c+\frac{1}{2}i)}\biggl \} \nonumber \\ +\sum_{n=0}^{\infty}\sum_{p=0}^{\infty}\frac{(a)_{n+2p+1}(b)_n(c-i)_{2p+1}~y^n~z^{2p+1}}{(d)_n(e)_{2p+1}~n!~(2p+1)!} \nonumber \\ \times \biggr \{ E''_i \frac{2^{2p+1}\Gamma(\frac{1}{2})\Gamma(-2p-c+i)\Gamma(1-c)}{\Gamma(1-c+\frac{1}{2}(i+|i|))\Gamma(-p+\frac{1}{2}i-\big [\frac{1+i}{2}\big ])\Gamma(-p+\frac{1}{2}-c+\frac{1}{2}i)}\nonumber \\ +F''_i\frac{2^{2p+1}\Gamma(\frac{1}{2})\Gamma(-2p-c+i)\Gamma(1-c)}{\Gamma(1-c+\frac{1}{2}(i+|i|))\Gamma(-p-\frac{1}{2}+\frac{1}{2}i-\big [\frac{i}{2}\big ])\Gamma(-p-c+\frac{1}{2}i)}\biggl \} \end{equation}$$ 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 ]:
(20) Γ ( α + n ) Γ ( α ) = ( α ) n , Γ ( α − n ) Γ ( α ) = ( − 1 ) n ( 1 − α ) n $$\begin{equation}\label{equation20} \frac{\Gamma(\alpha+n)}{\Gamma(\alpha)} = (\alpha)_n,~~\frac{\Gamma(\alpha-n)}{\Gamma(\alpha)}~=~\frac{(-1)^n}{(1-\alpha)_n} \end{equation}$$ (21) Γ 1 2 Γ ( 1 + α ) = 2 α Γ 1 2 + 1 2 α Γ 1 + 1 2 α $$\begin{equation}\Gamma\left (\frac{1}{2} \right )\Gamma(1+\alpha)~=~2^\alpha~\Gamma\left (\frac{1}{2}+\frac{1}{2}\alpha\right )\Gamma\left (1+\frac{1}{2}\alpha \right) \end{equation}$$ (22) ( α ) 2 n = 2 2 n 1 2 α n 1 2 α + 1 2 n $$\begin{equation}(\alpha)_{2n} = 2^{2n} \left (\frac{1}{2}\alpha \right)_n~\left (\frac{1}{2}\alpha+\frac{1}{2} \right)_n \end{equation}$$ (23) ( 2 n ) ! = 2 2 n 1 2 n n ! a n d ( 2 n + 1 ) ! = 2 2 n 3 2 n n ! . $$\begin{equation}\label{equation23} (2n)!~=~2^{2n} \left(\frac{1}{2}\right)_n n! ~~and~~ (2n+1)!~=~2^{2n} \left(\frac{3}{2}\right)_n n!. \end{equation}$$ 1. Taking i = 0 and d = b in (16), we get
(24) K 5 ( a , a , a , a ; b ′ , b ′ , b , b ; c ′ , c , b , b ; x , y , z , − z ) = X 4 ( a , b ′ ; b , c ′ , c ; − z 2 , x , y ) . $$\begin{equation}\label{equation24} K_5(a,a,a,a;b',b',b,b;c',c,b,b;x,y,z,-z)=X_4(a,b';b,c',c;-z^2,x,y). \end{equation}$$ 2. Taking i = 1 and d = b- 1 in (16), we get
(25) K 5 ( a , a , a , a ; b ′ , b ′ , b , b ; c ′ , c , b − 1 , b ; x , y , z , − z ) = X 4 ( a , b ′ ; b − 1 , c ′ , c ; − z 2 , x , y ) + a z b − 1 X 4 ( a + 1 , b ′ ; b , c ′ , c ; − z 2 , x , y ) . $$\begin{equation}
K_5(a,a,a,a;b',b',b,b;c',c,b-1,b;x,y,z,-z)\nonumber \\ =X_4(a,b';b-1,c',c;-z^2,x,y)+\frac{az}{b-1}X_4(a+1,b';b,c',c;-z^2,x,y). \end{equation}$$ 3. Taking i = - 1 and d = b in (16), we get
(26) K 5 ( a , a , a , a ; b ′ , b ′ , b , b ; c ′ , c , b , b − 1 ; x , y , z , − z ) = X 4 ( a , b ′ ; b − 1 , c ′ , c ; − z 2 , x , y ) − a z 2 X 4 ( a + 1 , b ′ ; b , c ′ , c ; − z 2 , x , y ) . $$\begin{equation} K_5(a,a,a,a;b',b',b,b;c',c,b,b-1;x,y,z,-z)\nonumber \\ =X_4(a,b';b-1,c',c;-z^2,x,y)-\frac{az}{2}X_4(a+1,b';b,c',c;-z^2,x,y). \end{equation}$$ 4. Taking i = 0 and e = 2c in (17), we get
(27) K 12 ( a , a , a , a ; b ′ , b , c , c ; d , d , 2 c , 2 c ; x , y , z , − z ) = X 7 ( a , b ′ , b ; d , c + 1 2 ; z 2 / 4 , x , y ) . $$\begin{equation}\label{equation27} K_{12}(a,a,a,a;b',b,c,c;d,d,2c,2c;x,y,z,-z) =X_7(a,b',b;d,c+\frac{1}{2};z^2/4,x,y). \end{equation}$$ 5. Taking i = 1 and e = 2c- 1 in (17), we get
(28) K 12 ( a , a , a , a ; b ′ , b , c − 1 , c ; d , d , 2 c − 1 , 2 c − 1 ; x , y , z , − z ) = X 7 ( a , b ′ , b ; d , c − 1 2 ; z 2 / 4 , x , y ) − a z 2 c − 1 X 7 ( a + 1 , b ′ , b ; d , c + 1 2 ; z 2 / 4 , x , y ) . $$\begin{equation} K_{12}(a,a,a,a;b',b,c-1,c;d,d,2c-1,2c-1;x,y,z,-z)\nonumber \\ =X_7(a,b',b;d,c-\frac{1}{2};z^2/4,x,y)-\frac{az}{2c-1}X_7(a+1,b',b;d,c+\frac{1}{2};z^2/4,x,y). \end{equation}$$ 6. Taking i = - 1 and e = 2c +1 in (17), we get
(29) K 12 ( a , a , a , a ; b ′ , b , c + 1 , c ; d , d , 2 c + 1 , 2 c + 1 ; x , y , z , − z ) = X 7 ( a , b ′ , b ; d , c + 1 2 ; z 2 / 4 , x , y ) − a z 2 c + 1 X 7 ( a + 1 , b ′ , b ; d , c + 3 2 ; z 2 / 4 , x , y ) . $$\begin{equation}K_{12}(a,a,a,a;b',b,c+1,c;d,d,2c+1,2c+1;x,y,z,-z)\nonumber \\ =X_7(a,b',b;d,c+\frac{1}{2};z^2/4,x,y)-\frac{az}{2c+1}X_7(a+1,b',b;d,c+\frac{3}{2};z^2/4,x,y). \end{equation}$$ 3 Applications to Generating Functions
Two interesting generating functions for Jacobi polynomials P n ( α , β ) ( x ) $P_n^{(\alpha,\beta)}(x)$ 7 ]
(30) ∑ n = 0 ∞ ( λ ) n ( − α − β ) n P n ( α − n , β − n ) ( x ) F 4 ( λ + n , γ ; δ , ρ ; y , z ) t n = [ 1 − ( 1 − x ) t 2 ] − λ × F E ( λ , λ , λ , − α , γ , γ ; − α − β , δ , ρ ; − 2 t 2 − ( 1 − x ) t , 2 y 2 − ( 1 − x ) t , 2 z 2 − ( 1 − x ) t ) $$\begin{equation}\sum_{n=0}^{\infty}\frac{(\lambda)_n}{(-\alpha-\beta)_n} P_n^{(\alpha-n,\beta-n)}(x)F_4(\lambda+n,\gamma;\delta,\rho;y,z)t^n = \bigg[1-\frac{(1-x)t}{2}\bigg]^{-\lambda} \nonumber \\ \times F_E \bigg(\lambda,\lambda,\lambda,-\alpha,\gamma,\gamma;-\alpha-\beta,\delta,\rho;\frac{-2t}{2-(1-x)t},\frac{2y}{2-(1-x)t},\frac{2z}{2-(1-x)t}\bigg) \end{equation}$$ and
(31) ∑ n = 0 ∞ ( λ ) n ( − α − β ) n P n ( α − n , β − n ) ( x ) F 1 ( λ + n , γ , ρ ; δ ; y , z ) t n = [ 1 − ( 1 − x ) t 2 ] − λ × F G ( λ , λ , λ , − α , γ , ρ ; − α − β , δ , δ ; − 2 t 2 − ( 1 − x ) t , 2 y 2 − ( 1 − x ) t , 2 z 2 − ( 1 − x ) t ) , $$\begin{equation}\label{equation31} \sum_{n=0}^{\infty}\frac{(\lambda)_n}{(-\alpha-\beta)_n} P_n^{(\alpha-n,\beta-n)}(x)F_1(\lambda+n,\gamma,\rho;\delta;y,z)t^n = \bigg[1-\frac{(1-x)t}{2}\bigg]^{-\lambda} \nonumber \\ \times F_G \bigg(\lambda,\lambda,\lambda,-\alpha,\gamma,\rho;-\alpha-\beta,\delta,\delta;\frac{-2t}{2-(1-x)t},\frac{2y}{2-(1-x)t},\frac{2z}{2-(1-x)t}\bigg), \end{equation}$$ where F 1 and F 4 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:
(32) ∑ n = 0 ∞ ( λ ) n ( − α − β ) n P n ( α − n , β − n ) ( x ) F 4 ( λ + n , γ ; δ , δ + i ; y , − y ) t n = [ 1 − ( 1 − x ) t 2 ] − λ ∑ n = 0 ∞ ∑ p = 0 ∞ ( λ ) n + 2 p ( − α ) n ( γ ) 2 p ( − α − β ) n ( δ ) 2 p n ! ( 2 p ) ! [ − 2 t 2 − ( 1 − x ) t ] n [ 2 y 2 − ( 1 − x ) t ] 2 p × { A i ′ 2 2 p Γ ( 1 2 ) Γ ( δ + i ) Γ ( δ + 2 p ) Γ ( δ + 2 p + 1 2 ( i + | i | ) ) Γ ( − p + 1 2 i + 1 2 − [ 1 + i 2 ] ) Γ ( p + δ + 1 2 i ) + B i ′ 2 2 p Γ ( 1 2 ) Γ ( δ + i ) Γ ( δ + 2 p ) Γ ( δ + 2 p + 1 2 ( i + | i | ) ) Γ ( − p + 1 2 i − [ i 2 ] ) Γ ( p + δ − 1 2 + 1 2 i ) } + ∑ n = 0 ∞ ∑ p = 0 ∞ ( λ ) n + 2 p + 1 ( − α ) n ( γ ) 2 p + 1 ( − α − β ) n ( δ ) 2 p + 1 n ! ( 2 p + 1 ) ! [ − 2 t 2 − ( 1 − x ) t ] n [ 2 y 2 − ( 1 − x ) t ] 2 p + 1 × { A i ″ 2 2 p + 1 Γ ( 1 2 ) Γ ( δ + i ) Γ ( δ + 2 p + 1 ) Γ ( δ + 2 p + 1 + 1 2 ( i + | i | ) ) Γ ( − p + 1 2 i − [ 1 + i 2 ] ) Γ ( p + 1 2 + δ + 1 2 i ) + B i ″ 2 2 p + 1 Γ ( 1 2 ) Γ ( δ + i ) Γ ( δ + 2 p + 1 ) Γ ( δ + 2 p + 1 + 1 2 ( i + | i | ) ) Γ ( − p − 1 2 + 1 2 i − [ i 2 ] ) Γ ( p + δ + 1 2 i ) } $$\begin{equation}\label{equation32} \sum_{n=0}^{\infty}\frac{(\lambda)_n}{(-\alpha-\beta)_n} P_n^{(\alpha-n,\beta-n)}(x)F_4(\lambda+n,\gamma;\delta,\delta+i;y,-y)t^n \nonumber \\ = \bigg[1-\frac{(1-x)t}{2}\bigg]^{-\lambda}\sum_{n=0}^{\infty}\sum_{p=0}^{\infty} \frac{(\lambda)_{n+2p}(-\alpha)_n(\gamma)_{2p}}{(-\alpha-\beta)_n(\delta)_{2p}~n!~(2p)!}\bigg[\frac{-2t}{2-(1-x)t}\bigg]^n\bigg[\frac{2y}{2-(1-x)t}\bigg]^{2p} \nonumber \\ \times \biggr \{ A'_i \frac{2^{2p}\Gamma(\frac{1}{2})\Gamma(\delta+i)\Gamma(\delta+2p)}{\Gamma(\delta+2p+\frac{1}{2}(i+|i|))\Gamma(-p+\frac{1}{2}i+\frac{1}{2}-\big [\frac{1+i}{2}\big ])\Gamma(p+\delta+\frac{1}{2}i)} \nonumber \\ +B'_i\frac{2^{2p}\Gamma(\frac{1}{2})\Gamma(\delta+i)\Gamma(\delta+2p)}{\Gamma(\delta+2p+\frac{1}{2}(i+|i|))\Gamma(-p+\frac{1}{2}i-\big [\frac{i}{2}\big ])\Gamma(p+\delta-\frac{1}{2}+\frac{1}{2}i)}\biggl \} \nonumber \\ +\sum_{n=0}^{\infty}\sum_{p=0}^{\infty}\frac{(\lambda)_{n+2p+1}(-\alpha)_n(\gamma)_{2p+1}}{(-\alpha-\beta)_n(\delta)_{2p+1}~n!~(2p+1)!}\bigg[\frac{-2t}{2-(1-x)t}\bigg]^n\bigg[\frac{2y}{2-(1-x)t}\bigg]^{2p+1} \nonumber \\ \times \biggr \{ A''_i \frac{2^{2p+1}\Gamma(\frac{1}{2})\Gamma(\delta+i)\Gamma(\delta+2p+1)}{\Gamma(\delta+2p+1+\frac{1}{2}(i+|i|))\Gamma(-p+\frac{1}{2}i-\big [\frac{1+i}{2}\big ])\Gamma(p+\frac{1}{2}+\delta+\frac{1}{2}i)}\nonumber \\ +B''_i\frac{2^{2p+1}\Gamma(\frac{1}{2})\Gamma(\delta+i)\Gamma(\delta+2p+1)}{\Gamma(\delta+2p+1+\frac{1}{2}(i+|i|))\Gamma(-p-\frac{1}{2}+\frac{1}{2}i-\big [\frac{i}{2}\big ])\Gamma(p+\delta+\frac{1}{2}i)} \biggl \} \end{equation}$$ 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:
(33) ∑ n = 0 ∞ ( λ ) n ( − α − β ) n P n ( α − n , β − n ) ( x ) F 1 ( λ + n , ρ − i , ρ ; δ ; y , − y ) t n = [ 1 − ( 1 − x ) t 2 ] − λ ∑ n = 0 ∞ ∑ p = 0 ∞ ( λ ) n + 2 p ( − α ) n ( ρ − i ) 2 p ( − α − β ) n ( δ ) 2 p n ! ( 2 p ) ! [ − 2 t 2 − ( 1 − x ) t ] n [ 2 y 2 − ( 1 − x ) t ] 2 p × { E i ′ 2 2 p Γ ( 1 2 ) Γ ( 1 − 2 p − ρ + i ) Γ ( 1 − ρ ) Γ ( 1 − ρ + 1 2 ( i + | i | ) ) Γ ( − p + 1 2 i + 1 2 − [ 1 + i 2 ] ) Γ ( 1 − p − ρ + 1 2 i ) + F i ′ 2 2 p Γ ( 1 2 ) Γ ( 1 − 2 p − ρ + i ) Γ ( 1 − ρ ) Γ ( 1 − ρ + 1 2 ( i + | i | ) ) Γ ( − p + 1 2 i − [ i 2 ] ) Γ ( − p + 1 2 − ρ + 1 2 i ) } + ∑ n = 0 ∞ ∑ p = 0 ∞ ( λ ) n + 2 p + 1 ( − α ) n ( ρ − i ) 2 p + 1 ( − α − β ) n ( δ ) 2 p + 1 n ! ( 2 p + 1 ) ! [ − 2 t 2 − ( 1 − x ) t ] n [ 2 y 2 − ( 1 − x ) t ] 2 p + 1 × { E i ″ 2 2 p + 1 Γ ( 1 2 ) Γ ( − 2 p − ρ + i ) Γ ( 1 − ρ ) Γ ( 1 − ρ + 1 2 ( i + | i | ) ) Γ ( − p + 1 2 i − [ 1 + i 2 ] ) Γ ( − p + 1 2 − ρ + 1 2 i ) + F i ″ 2 2 p + 1 Γ ( 1 2 ) Γ ( − 2 p − ρ + i ) Γ ( 1 − ρ ) Γ ( 1 − ρ + 1 2 ( i + | i | ) ) Γ ( − p − 1 2 + 1 2 i − [ i 2 ] ) Γ ( − p − ρ + 1 2 i ) } $$\begin{equation}\sum_{n=0}^{\infty}\frac{(\lambda)_n}{(-\alpha-\beta)_n} P_n^{(\alpha-n,\beta-n)}(x)F_1(\lambda+n,\rho-i,\rho;\delta;y,-y)t^n \nonumber \\ = \bigg[1-\frac{(1-x)t}{2}\bigg]^{-\lambda}\sum_{n=0}^{\infty}\sum_{p=0}^{\infty} \frac{(\lambda)_{n+2p}(-\alpha)_n(\rho-i)_{2p}}{(-\alpha-\beta)_n(\delta)_{2p}~n!~(2p)!}\bigg[\frac{-2t}{2-(1-x)t}\bigg]^n\bigg[\frac{2y}{2-(1-x)t}\bigg]^{2p} \nonumber \\ \times \biggr \{ E'_i \frac{2^{2p}\Gamma(\frac{1}{2})\Gamma(1-2p-\rho+i)\Gamma(1-\rho)}{\Gamma(1-\rho+\frac{1}{2}(i+|i|))\Gamma(-p+\frac{1}{2}i+\frac{1}{2}-\big [\frac{1+i}{2}\big ])\Gamma(1-p-\rho+\frac{1}{2}i)} \nonumber \\ +F'_i\frac{2^{2p}\Gamma(\frac{1}{2})\Gamma(1-2p-\rho+i)\Gamma(1-\rho)}{\Gamma(1-\rho+\frac{1}{2}(i+|i|))\Gamma(-p+\frac{1}{2}i-\big [\frac{i}{2}\big ])\Gamma(-p+\frac{1}{2}-\rho+\frac{1}{2}i)}\biggl \} \nonumber \\ +\sum_{n=0}^{\infty}\sum_{p=0}^{\infty}\frac{(\lambda)_{n+2p+1}(-\alpha)_n(\rho-i)_{2p+1}}{(-\alpha-\beta)_n(\delta)_{2p+1}~n!~(2p+1)!}\bigg[\frac{-2t}{2-(1-x)t}\bigg]^n\bigg[\frac{2y}{2-(1-x)t}\bigg]^{2p+1} \nonumber \\ \times \biggr \{ E''_i \frac{2^{2p+1}\Gamma(\frac{1}{2})\Gamma(-2p-\rho+i)\Gamma(1-\rho)}{\Gamma(1-\rho+\frac{1}{2}(i+|i|))\Gamma(-p+\frac{1}{2}i-\big [\frac{1+i}{2}\big ])\Gamma(-p+\frac{1}{2}-\rho+\frac{1}{2}i)}\nonumber \\ +F''_i\frac{2^{2p+1}\Gamma(\frac{1}{2})\Gamma(-2p-\rho+i)\Gamma(1-\rho)}{\Gamma(1-\rho+\frac{1}{2}(i+|i|))\Gamma(-p-\frac{1}{2}+\frac{1}{2}i-\big [\frac{i}{2}\big ])\Gamma(-p-\rho+\frac{1}{2}i)} \biggl \} \end{equation}$$ 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).
1. Taking i = 0 in (32) and (33), we get
(34) ∑ n = 0 ∞ ( λ ) n ( − α − β ) n P n ( α − n , β − n ) ( x ) F 4 ( λ + n , γ ; δ , δ ; y , − y ) t n = ( 1 − ( 1 − x ) t 2 ) − λ X 1 : 2 ; 1 0 : 3 ; 1 λ : 1 2 γ , 1 2 γ + 1 2 ; − α ; − : δ , 1 2 δ , 1 2 δ + 1 2 ; − α − β ; − ( 2 y 2 − ( 1 − x ) t ) 2 , − 2 t 2 − ( 1 − x ) t $$\begin{equation}\label{equation34} \sum_{n=0}^{\infty}\frac{(\lambda)_n}{(-\alpha-\beta)_n} P_n^{(\alpha-n,\beta-n)}(x)F_4(\lambda+n,\gamma;\delta,\delta;y,-y)t^n \nonumber \\ =\bigg(1-\frac{(1-x)t}{2}\bigg)^{-\lambda} X \begin{array}{c} 1:2;1 \\ 0:3;1 \end{array} \left[ \begin{array}{cccccc} \lambda&:&\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}&;&-\alpha&; \\ \\ -&:&\delta,\frac{1}{2}\delta,\frac{1}{2}\delta+\frac{1}{2}&;&-\alpha-\beta&; \end{array} \begin{array}{c} -\bigg(\frac{2y}{2-(1-x)t}\bigg)^2,\frac{-2t}{2-(1-x)t} \end{array} \right] \end{equation}$$ and
(35) ∑ n = 0 ∞ ( λ ) n ( − α − β ) n P n ( α − n , β − n ) ( x ) F 1 ( λ + n , ρ , ρ ; δ ; y , − y ) t n = ( 1 − ( 1 − x ) t 2 ) − λ X 1 : 1 ; 1 0 : 2 ; 1 λ : ρ ; − α ; − : 1 2 δ , 1 2 δ + 1 2 ; − α − β ; ( y 2 − ( 1 − x ) t ) 2 , − 2 t 2 − ( 1 − x ) t . $$\begin{equation}\label{equation35} \sum_{n=0}^{\infty}\frac{(\lambda)_n}{(-\alpha-\beta)_n} P_n^{(\alpha-n,\beta-n)}(x)F_1(\lambda+n,\rho,\rho;\delta;y,-y)t^n\nonumber \\ =\bigg(1-\frac{(1-x)t}{2}\bigg)^{-\lambda} X \begin{array}{c} 1:1;1 \\ 0:2;1 \end{array} \left[ \begin{array}{cccccc} \lambda&:&\rho&;&-\alpha&; \\ \\ -&:&\frac{1}{2}\delta,\frac{1}{2}\delta+\frac{1}{2}&;&-\alpha-\beta&; \end{array} \begin{array}{c} \bigg(\frac{y}{2-(1-x)t}\bigg)^2,\frac{-2t}{2-(1-x)t} \end{array} \right]. \end{equation}$$ 2. Taking i = 1 in (32) and (33), we get
(36) ∑ n = 0 ∞ ( λ ) n ( − α − β ) n P n ( α − n , β − n ) ( x ) F 4 ( λ + n , γ ; δ , δ + 1 ; y , − y ) t n = ( 1 − ( 1 − x ) t 2 ) − λ { X 1 : 2 ; 1 0 : 3 ; 1 λ : 1 2 γ , 1 2 γ + 1 2 ; − α ; − : δ , 1 2 δ + 1 2 , 1 2 δ + 1 ; − α − β ; − ( 2 y 2 − ( 1 − x ) t ) 2 , − 2 t 2 − ( 1 − x ) t + 2 λ γ y δ ( δ + 1 ) ( 2 − t + x t ) × X 1 : 2 ; 1 0 : 3 ; 1 λ + 1 : 1 2 γ + 1 2 , 1 2 γ + 1 ; − α ; − : δ + 1 , 1 2 δ + 1 , 1 2 δ + 3 2 ; − α − β ; − ( 2 y 2 − ( 1 − x ) t ) 2 , − 2 t 2 − ( 1 − x ) t } $$\begin{align*}
&\sum_{n=0}^{\infty}\frac{(\lambda)_n}{(-\alpha-\beta)_n} P_n^{(\alpha-n,\beta-n)}(x)F_4(\lambda+n,\gamma;\delta,\delta+1;y,-y)t^n
\\
&=\bigg(1-\frac{(1-x)t}{2}\bigg)^{-\lambda} \Bigg\{ X \begin{array}{c} 1:2;1 \\ 0:3;1 \end{array} \left[ \begin{array}{cccccc} \lambda&:&\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}&;&-\alpha&; \\ \\ -&:&\delta,\frac{1}{2}\delta+\frac{1}{2},\frac{1}{2}\delta+1&;&-\alpha-\beta&; \end{array} \begin{array}{c} -\bigg(\frac{2y}{2-(1-x)t}\bigg)^2,\frac{-2t}{2-(1-x)t} \end{array} \right] \\ &+\frac{2\lambda \gamma y}{\delta(\delta+1)(2-t+xt)}
\times X \begin{array}{c} 1:2;1 \\ 0:3;1 \end{array} \left[ \begin{array}{cccccc} \lambda+1&:&\frac{1}{2}\gamma+\frac{1}{2},\frac{1}{2}\gamma+1&;&-\alpha&; \\ \\ -&:&\delta+1,\frac{1}{2}\delta+1,\frac{1}{2}\delta+\frac{3}{2}&;&-\alpha-\beta&; \end{array} \begin{array}{c} -\bigg(\frac{2y}{2-(1-x)t}\bigg)^2,\frac{-2t}{2-(1-x)t} \end{array} \right] \Bigg\}
\end{align*}$$
and
(37) ∑ n = 0 ∞ ( λ ) n ( − α − β ) n P n ( α − n , β − n ) ( x ) F 1 ( λ + n , ρ − 1 , ρ ; δ ; y , − y ) t n = ( 1 − ( 1 − x ) t 2 ) − λ { X 1 : 1 ; 1 0 : 2 ; 1 λ : ρ ; − α ; − : 1 2 δ , 1 2 δ + 1 2 ; − α − β ; ( y 2 − ( 1 − x ) t ) 2 , − 2 t 2 − ( 1 − x ) t + 2 λ y δ ( 2 − t + x t ) X 1 : 1 ; 1 0 : 2 ; 1 λ + 1 : ρ ; − α ; − : 1 2 δ + 1 2 , 1 2 δ + 1 ; − α − β ; ( y 2 − ( 1 − x ) t ) 2 , − 2 t 2 − ( 1 − x ) t } . $$\begin{equation}\label{equation37} \sum_{n=0}^{\infty}\frac{(\lambda)_n}{(-\alpha-\beta)_n} P_n^{(\alpha-n,\beta-n)}(x)F_1(\lambda+n,\rho-1,\rho;\delta;y,-y)t^n \nonumber \\ =\bigg(1-\frac{(1-x)t}{2}\bigg)^{-\lambda} \Bigg\{ X \begin{array}{c} 1:1;1 \\ 0:2;1 \end{array} \left[ \begin{array}{cccccc} \lambda&:&\rho&;&-\alpha&; \\ \\ -&:&\frac{1}{2}\delta,\frac{1}{2}\delta+\frac{1}{2}&;&-\alpha-\beta&; \end{array} \begin{array}{c} \bigg(\frac{y}{2-(1-x)t}\bigg)^2,\frac{-2t}{2-(1-x)t} \end{array} \right] \nonumber \\ +\frac{2\lambda y}{\delta(2-t+xt)} X \begin{array}{c} 1:1;1 \\ 0:2;1 \end{array} \left[ \begin{array}{cccccc} \lambda+1&:&\rho&;&-\alpha&; \\ \\ -&:&\frac{1}{2}\delta+\frac{1}{2},\frac{1}{2}\delta+1&;&-\alpha-\beta&; \end{array} \begin{array}{c} \bigg(\frac{y}{2-(1-x)t}\bigg)^2,\frac{-2t}{2-(1-x)t} \end{array} \right] \Bigg\}. \end{equation}$$ 3. Taking i = - 1 in (32) and (33), we get
(38) ∑ n = 0 ∞ ( λ ) n ( − α − β ) n P n ( α − n , β − n ) ( x ) F 4 ( λ + n , γ ; δ , δ − 1 ; y , − y ) t n = ( 1 − ( 1 − x ) t 2 ) − λ { X 1 : 2 ; 1 0 : 3 ; 1 λ : 1 2 γ , 1 2 γ + 1 2 ; − α ; − : δ − 1 , 1 2 δ , 1 2 δ + 1 2 ; − α − β ; − ( 2 y 2 − ( 1 − x ) t ) 2 , − 2 t 2 − ( 1 − x ) t − λ γ y δ ( 2 − t + x t ) X 1 : 2 ; 1 0 : 3 ; 1 λ + 1 : 1 2 γ + 1 2 , 1 2 γ + 1 ; − α ; − : δ , 1 2 δ + 1 2 , δ + 1 ; − α − β ; − ( 2 y 2 − ( 1 − x ) t ) 2 , − 2 t 2 − ( 1 − x ) t } $$\begin{equation} \sum_{n=0}^{\infty}\frac{(\lambda)_n}{(-\alpha-\beta)_n} P_n^{(\alpha-n,\beta-n)}(x)F_4(\lambda+n,\gamma;\delta,\delta-1;y,-y)t^n\nonumber \\ =\bigg(1-\frac{(1-x)t}{2}\bigg)^{-\lambda} \Bigg\{ X \begin{array}{c} 1:2;1 \\ 0:3;1 \end{array} \left[ \begin{array}{cccccc} \lambda&:&\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}&;&-\alpha&; \\ \\ -&:&\delta-1,\frac{1}{2}\delta,\frac{1}{2}\delta+\frac{1}{2}&;&-\alpha-\beta&; \end{array} \begin{array}{c} -\bigg(\frac{2y}{2-(1-x)t}\bigg)^2,\frac{-2t}{2-(1-x)t} \end{array} \right] \nonumber \\ -\frac{\lambda \gamma y}{\delta(2-t+xt)} X \begin{array}{c} 1:2;1 \\ 0:3;1 \end{array} \left[ \begin{array}{cccccc} \lambda+1&:&\frac{1}{2}\gamma+\frac{1}{2},\frac{1}{2}\gamma+1&;&-\alpha&; \\ \\ -&:&\delta,\frac{1}{2}\delta+\frac{1}{2},\delta+1&;&-\alpha-\beta&; \end{array} \begin{array}{c} -\bigg(\frac{2y}{2-(1-x)t}\bigg)^2,\frac{-2t}{2-(1-x)t} \end{array} \right] \Bigg\} \end{equation}$$ and
(39) ∑ n = 0 ∞ ( λ ) n ( − α − β ) n P n ( α − n , β − n ) ( x ) F 1 ( λ + n , ρ + 1 , ρ ; δ ; y , − y ) t n = ( 1 − ( 1 − x ) t 2 ) − λ { X 1 : 1 ; 1 0 : 2 ; 1 λ : ρ + 1 ; − α ; − : 1 2 δ , 1 2 δ + 1 2 ; − α − β ; ( y 2 − ( 1 − x ) t ) 2 , − 2 t 2 − ( 1 − x ) t + 2 λ y δ ( 2 − t + x t ) X 1 : 1 ; 1 0 : 2 ; 1 λ + 1 : ρ + 1 ; − α ; − : 1 2 δ + 1 2 , 1 2 δ + 1 ; − α − β ; ( y 2 − ( 1 − x ) t ) 2 , − 2 t 2 − ( 1 − x ) t } . $$\begin{equation} \sum_{n=0}^{\infty}\frac{(\lambda)_n}{(-\alpha-\beta)_n} P_n^{(\alpha-n,\beta-n)}(x)F_1(\lambda+n,\rho+1,\rho;\delta;y,-y)t^n \nonumber \\ =\bigg(1-\frac{(1-x)t}{2}\bigg)^{-\lambda} \Bigg\{ X \begin{array}{c} 1:1;1 \\ 0:2;1 \end{array} \left[ \begin{array}{cccccc} \lambda&:&\rho+1&;&-\alpha&; \\ \\ -&:&\frac{1}{2}\delta,\frac{1}{2}\delta+\frac{1}{2}&;&-\alpha-\beta&; \end{array} \begin{array}{c} \bigg(\frac{y}{2-(1-x)t}\bigg)^2,\frac{-2t}{2-(1-x)t} \end{array} \right] \nonumber \\ +\frac{2\lambda y}{\delta(2-t+xt)} X \begin{array}{c} 1:1;1 \\ 0:2;1 \end{array} \left[ \begin{array}{cccccc} \lambda+1&:&\rho+1&;&-\alpha&; \\ \\ -&:&\frac{1}{2}\delta+\frac{1}{2},\frac{1}{2}\delta+1&;&-\alpha-\beta&; \end{array} \begin{array}{c} \bigg(\frac{y}{2-(1-x)t}\bigg)^2,\frac{-2t}{2-(1-x)t} \end{array} \right] \Bigg\}. \end{equation}$$ The other special cases of (32) and (33) can also be obtained in the similar manner.
