General integral formulas involving Humbert hypergeometric functions

Mathematica Moravica Vol. 21, No. 2 (2017), 75–84

General integral formulas involving Humbert hypergeometric functions of two variables Ahmed Ali Atash and Hussein Saleh Bellehaj Abstract. In this paper, we have established two general integral formulas involving Humbert hypergeometric functions of two variables Φ2 and Ψ2 . The results are obtained with the help of a generalization of classical Kummer’s summation theorem on the sum of the series 2 F1 (−1) due to Lavoie et al. [5]. Some interesting applications are also presented.

1. Introduction The confluent hypergeometric functions Φ2 and Ψ2 are defined and represented as follows [2,8]: (1.1)

∞ X (a)m (a0 )n xm y n Φ2 [a,a ;b;x;y]= (b)m+n m! n! 0

m,n=0

and (1.2)

Ψ2 [a;b,b0 ;x;y]=

∞ X (a)m+n xm y n , (b)m (b0 )n m! n!

m,n=0

where (λ)n denotes the Pochhammer symbol defined by [8] ( 1, if n=0 (1.3) (λ)n = λ(λ+1)...(λ+n−1), if n=1,2,3,... Exton [3,4] gave the definitions and the Laplace integral representations of the quadruple hypergeometric functions K5 and K12 as follows: 2010 Mathematics Subject Classification. 33C05, 33C65, 33C70. Key words and phrases. Humbert functions, Exton functions, Kampé de Fériet function, Integral formulas, Kummer’s theorem. 75

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2017 Mathematica Moravica

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General integral formulas involving Humbert hypergeometric. . .

K5 (a,a,a,a;b1 ,b1 ,b2 ,b2 ;c1 ,c2 ,c3 ,c4 ;x,y,z,t) (1.4)

∞ X (a)p+q+r+s (b1 )p+q (b2 )r+s xp y q z r ts (c1 )p (c2 )q (c3 )r (c4 )s p!q!r!s! p,q,r,s=0 Z ∞ 1 = e−s sa−1 Ψ2 (b1 ;c1 ,c2 ;xs,ys)Ψ2 (b2 ;c3 ,c4 ;zs,ts)ds Γ(a) 0

=

and K12 (a,a,a,a;b1 ,b2 ,b3 ,b4 ;c1 ,c1 ,c2 ,c2 ;x,y,z,t) (1.5)

∞ X (a)p+q+r+s (b1 )p (b2 )q (b3 )r (b4 )s xp y q z r ts (c1 )p+q (c2 )r+s p!q!r!s! p,q,r,s=0 Z ∞ 1 e−s sa−1 Φ2 (b1 ,b2 ;c1 ;xs,ys)Φ2 (b3 ,b4 ;c2 ;zs,ts)ds. = Γ(a) 0

=

p:q;k The Kampé de Fériet function of two variables Fl:m;n [x,y] is defined and represented as follows [8]:   (ap ) : (bq ) ; (ck ) ; p:q;k  x , y Fl:m;n (αl ) : (βm ) ; (γn ) ; Qp Q Q ∞ X j=1 (aj )r+s qj=1 (bj )r kj=1 (cj )s xr y s (1.6) = . Ql Qm Qn r! s! j=1 (αj )r+s j=1 (βj )r j=1 (γj )s r,s=0

In the present investigation, we shall require the following generalization of the classical Kummer’s theorem for the series 2 F1 (−1) [5]:   Γ( 21 )Γ(1+a−b+i)Γ(1−b) a,b ; −1 = 2 F1 1+a−b+i; 2a Γ(1−b+ 21 (i+|i|)) Ai ×{ 1 1 1 1+i Γ( 2 a+ 2 i+ 2 −[ 2 ])Γ(1+ 21 a−b+ 12 i) Bi + 1 } Γ( 2 a+ 21 i−[ 2i ])Γ( 12 + 12 a−b+ 21 i) 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 [5]. When i=0, (1.7) reduces immediately to the classical Kummer’s theorem [1],(see also [6])   Γ(1+a−b)Γ( 21 ) a,b ; −1 (1.7) F = . 2 1 1+a−b; 2a Γ(1+ 21 a−b)Γ( 12 a+ 21 )

Ahmed Ali Atash and Hussein Saleh Bellehaj

The following results will be required also [8]: Γ(α+n) Γ(α−n) (1−)n , , = Γ(α) Γ(α) (1−α)n       1 1 1 1 Γ Γ(1+α) = 2α Γ + α Γ 1+ α , 2 2 2 2     1 1 2n 1 α α+ (α)2n =2 2 n 2 2 n (α)n =

(1.8)

(1.9)

(1.10) and

    1 2n 3 (2n)! = 2 n!, (2n+1)! = 2 n!. 2 n 2 n 2n

(1.11)

2. Main Integrals Formulas First Integral Z ∞ 1 e−s sa−1 Ψ2 (b1 ;c1 ,c1 +i;xs,−xs)Ψ2 (b2 ;c2 ,c2 +i;ys,−ys)ds Γ(a) 0 =

∞ ∞ X X (a)2m1 +2m2 (b1 )2m1 (b2 )2m2 x2m1 y 2m2 (c1 )2m1 (c2 )2m2 (2m1 )!(2m2 )!

m1 =0 m2 =0

(1)

(1)

(2)

(2)

×(Ai C1 +Bi D1 )(Ai C2 +Bi D2 ) ∞ ∞ X X (a)2m1 +2m2 +1 (b1 )2m1 +1 (b2 )2m2 x2m1 +1 y 2m2 (c1 )2m1 +1 (c2 )2m2 (2m1 +1)!(2m2 )!

+

m1 =0 m2 =0

(3)

×(Ai

(3)

E1 +Bi

(2)

F1 )(Ai

(2)

C2 +Bi

D2 )

∞ ∞ X X (a)2m1 +2m2 +1 (b1 )2m1 (b2 )2m2 +1 x2m1 y 2m2 +1 + (c1 )2m1 (c2 )2m2 +1 (2m1 )!(2m2 +1)! m1 =0 m2 =0

(1)

×(Ai

(1)

C1 +Bi

(4)

D1 )(Ai

(4)

E2 +Bi

F2 )

∞ ∞ X X (a)2m1 +2m2 +2 (b1 )2m1 +1 (b2 )2m2 +1 x2m1 +1 y 2m2 +1 + (c1 )2m1 +1 (c2 )2m2 +1 (2m1 +1)!(2m2 +1)! m1 =0 m2 =0

(3)

×(Ai

(2.1)

(3)

E1 +Bi

(4)

F1 )(Ai

(4)

E2 +Bi

F2 ),

where Cr =

22mr Γ( 21 )Γ(cr +i)Γ(cr +2mr ) 1 Γ(cr +2mr + 21 (i+|i|))Γ(−mr + 12 i+ 21 −[ 1+i 2 ])Γ(mr +cr + 2 i)

77

78

General integral formulas involving Humbert hypergeometric. . .

Dr = Er =

22mr Γ( 21 )Γ(cr +i)Γ(cr +2mr ) Γ(cr +2mr + 12 (i+|i|))Γ(−mr + 21 i−[ 2i ])Γ(mr +cr − 21 + 12 i) 22mr +1 Γ( 12 )Γ(cr +i)Γ(cr +2mr +1)

1 1 Γ(cr +2mr +1+ 12 (i+|i|))Γ(−mr + 21 i−[ 1+i 2 ])Γ(mr + 2 +cr + 2 i)

22mr +1 Γ( 12 )Γ(cr +i)Γ(cr +2mr +1)

Fr =

Γ(cr +2mr +1+ 21 (i+|i|))Γ(−mr − 12 + 21 i−[ 2i ])Γ(mr +cr + 12 i) (for i=0,±1,±2,±3,±4,±5 ; r=1,2). (1)

(1)

The coefficients Ai and Bi can be obtained from the tables of Ai and Bi given in [5] by replacing a and b by −2m1 and 1−c1 −2m1 , the coefficients (2) (2) Ai and Bi can be obtained from the tables of Ai and Bi by replacing a (3) (3) and b by −2m2 and 1−c2 −2m2 , the coefficients Ai and Bi can be obtained from the tables of Ai and Bi by replacing a and b by −2m1 −1 and (4) (4) −c1 −2m1 and the coefficients Ai and Bi can be obtained from the tables of Ai and Bi by replacing a and b by −2m2 −1 and −c2 −2m2 . Second Integral Z ∞ 1 e−s sa−1 Φ2 (b1 −i,b1 ;c1 ;xs,−xs)Φ2 (b2 −i,b2;c2 ;ys,−ys)ds Γ(a) 0 =

∞ ∞ X X (a)2m1 +2m2 (b1 −i)2m1 (b2 −i)2m2 x2m1 y 2m2 (c1 )2m1 (c2 )2m2 (2m1 )!(2m2 )!

m1 =0 m2 =0

0(1)

0(1)

×(Ai C10 +Bi

0(2)

0(2)

D10 )(Ai C20 +Bi

D20 )

∞ ∞ X X (a)2m1 +2m2 +1 (b1 −i)2m1 +1 (b2 −i)2m2 x2m1 +1 y 2m2 + (c1 )2m1 +1 (c2 )2m2 (2m1 +1)!(2m2 )! m1 =0 m2 =0

0(3)

0(3)

×(Ai E10 +Bi

0(2)

0(2)

F10 )(Ai C20 +Bi

D20 )

∞ ∞ X X (a)2m1 +2m2 +1 (b1 −i)2m1 (b2 −i)2m2 +1 x2m1 y 2m2 +1 + (c1 )2m1 (c2 )2m2 +1 (2m1 )!(2m2 +1)! m1 =0 m2 =0

0(1)

0(1)

×(Ai C10 +Bi

0(4)

0(4)

D10 )(Ai E20 +Bi

F20 )

∞ ∞ X X (a)2m1 +2m2 +2 (b1 −i)2m1 +1 (b2 −i)2m2 +1 x2m1 +1 y 2m2 +1 + (c1 )2m1 +1 (c2 )2m2 +1 (2m1 +1)!(2m2 +1)! m1 =0 m2 =0

(2.2)

0(3)

0(3)

× (Ai E10 +Bi

0(4)

0(4)

F10 )(Ai E20 +Bi

F20 ),

Ahmed Ali Atash and Hussein Saleh Bellehaj

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where Cr0 =

22mr Γ( 21 )Γ(1−2mr −br +i)Γ(1−br ) 1 Γ(1−br + 12 (i+|i|))Γ(−mr + 21 i+ 21 −[ 1+i 2 ])Γ(1−mr −br + 2 i)

Dr0 = Er0 = Fr0 =

22mr Γ( 21 )Γ(1−2mr −br +i)Γ(1−br ) Γ(1−br + 12 (i+|i|))Γ(−mr + 21 i−[ 2i ])Γ(−mr + 12 −br + 21 i) 22mr +1 Γ( 12 )Γ(−2mr −br +i)Γ(1−br )

1 1 Γ(1−br + 12 (i+|i|))Γ(−mr + 21 i−[ 1+i 2 ])Γ(−mr + 2 −br + 2 i)

22mr +1 Γ( 21 )Γ(−2mr −br +i)Γ(1−br ) Γ(1−br + 12 (i+|i|))Γ(−mr − 21 + 12 i−[ 2i ])Γ(−mr −br + 21 i)

for i=0,±1,±2,±3,±4,±5 ; r=1,2. 0(1) 0(1) The coefficients Ai and Bi can be obtained from the tables of Ai 0(2) 0(2) and Bi given in [5] by replacing a by −2m1 , the coefficients Ai and Bi can be obtained from the tables of Ai and Bi by replacing a by −2m2 , the 0(3) 0(3) coefficients Ai and Bi can be obtained from the tables of Ai and Bi by 0(4) 0(4) replacing a by −2m1 −1 and the coefficients Ai and Bi can be obtained from the tables of Ai and Bi by replacing a by −2m2 −1. Proof of the first integral: Denoting the left hand side of (2.1) by I, then from the definition (1.4), we have (2.3) ∞ X (a)m1 +p1 +m2 +p2 (b1 )m1 +p1 (b2 )m2 +p2 xm1 (−x)p1 y m2 (−y)p2 . I= (c1 )m1 (c1 +i)p1 (c2 )m2 (c2 +i)p2 m1 !p1 !m2 !p2 ! m1 ,p1 ,m2 ,p2 =0

Now, using the well-known results [8] (α)m+n = (α)m (α+m)n ∞ X ∞ X

A(n,m) =

m=0n=0

(α)m−n =

∞ X m X

A(n,m−n),

m=0n=0 n (−1) (α)m

, 0≤n≤m (1−α−m)n (−1)n m! (m−n)!= , 0≤n≤m, (−m)n then after a little simplification, we have ∞ X ∞ X (a)m1 +m2 (b1 )m1 (b2 )m2 xm1 y m2 I= (c1 )m1 (c2 )m2 m1 !m2 ! m1 =0m2 =0

80

General integral formulas involving Humbert hypergeometric. . .

    −m1 , 1−c1 −m1 ; −m2 , 1−c2 −m2 ; −12 F1  −1. 2 F1  c1 +i ; c2 +i ; Separating into its even and odd terms, we have I=

∞ X ∞ X (a)2m1 +2m2 (b1 )2m1 (b2 )2m2 x2m1 y 2m2 (c1 )2m1 (c2 )2m2 (2m1 )!(2m2 )!

m1 =0m2 =0

    −2m1 ,1−c1 −2m1 ; −2m2 ,1−c2 −2m2 ; −12 F1  −1 ×2 F1  c1 +i ; c2 +i ;

+

∞ ∞ X X (a)2m1 +2m2 +1 (b1 )2m1 +1 (b2 )2m2 x2m1 +1 y 2m2 (c1 )2m1 +1 (c2 )2m2 (2m1 +1)!(2m2 )!

m1 =0m2 =0

    −2m1 −1,−c1 −2m1 ; −2m2 ,1−c2 −2m2 ; −12 F1  −1 ×2 F1  c1 +i ; c2 +i ;

+

∞ X ∞ X (a)2m1 +2m2 +1 (b1 )2m1 (b2 )2m2 +1 x2m1 y 2m2 +1 (c1 )2m1 (c2 )2m2 +1 (2m1 )!(2m2 +1)!

m1 =0m2 =0

    −2m1 ,1−c1 −2m1 ; −2m2 −1,−c2 −2m2 ; −12 F1  −1 ×2 F1  c1 +i ; c2 +i ;

+

∞ X ∞ X (a)2m1 +2m2 +2 (b1 )2m1 +1 (b2 )2m2 +1 x2m1 +1 y 2m2 +1 (c1 )2m1 +1 (c2 )2m2 +1 (2m1 +1)!(2m2 +1)!

m1 =0m2 =0

    −2m1 −1,−c1 −2m1 ; −2m2 −1,−c2 −2m2 ; −12 F1  −1. ×2 F1  c1 +i ; c2 +i ; Finally , if we use the generalized Kummer’s theorem (1.7), then, after a little simplification, we readily arrive at the right hand side of (2.1).This completes the proof of the first integral . The proof of the second integral is similar to that of the first integral with the only difference that we use here the result (1.5).

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3. Applications (i) In (2.1), if we take i=0 and use the results (1.9)–(1.12), we get after a little simplification the following integral formula: Z ∞ 1 e−s sa−1 Ψ2 (b1 ;c1 ,c1 ;xs,−xs)Ψ2 (b2 ;c2 ,c2 ;ys,−ys)ds Γ(a) 0 1 1  (3.1) a, 2 a+ 21 : 12 b1 , 21 b1 + 21 ; 12 b2 , 12 b2 + 21 ; 2 2:2;2 −4x2 ,−4y 2 . =F 0:3;3 1 1 1 1 1 1 − :c1 , 2 c1 , 2 c1 + 2 ;c2 , 2 c2 , 2 c2 + 2 ; Now, in (3.1) using the result [7] 

1 1 1 2 a, 2 a+ 2

;



−x2 , Ψ2 (a;b,b;x,−x)=2 F3  1 1 1 b, 2 b, 2 b+ 2 ;

(3.2) we get

(3.3)



1 1 1 2 b1 , 2 b1 + 2

 ; 1 −(xs)2  e−s sa−1 2 F3  Γ(a) 0 c1 , 12 c1 , 21 c1 + 12 ;   1 1 1 b , b + ; 2 2 2 2 2 −(ys)2 ds ×2 F3  1 1 1 c2 , 2 c2 , 2 c2 + 2 ; 1 1  a, 2 a+ 21 : 12 b1 , 21 b1 + 21 ; 12 b2 , 12 b2 + 21 ; 2 2:2;2 −4x2 ,−4y 2 , =F 0:3;3 1 1 1 1 1 1 − :c1 , 2 c1 , 2 c1 + 2 ;c2 , 2 c2 , 2 c2 + 2 ; Z

∞

which for c1 =b1 , c2 =b2 reduces to the following integral in terms of Appell function F4 [8] Z ∞ 1 e−s sa−1 0 F1 (−;b1 ;−x2 s2 )0 F1 (−;b2 ;−y 2 s2 )ds Γ(a) 0 (3.4) 1 1 1 =F4 [ a, a+ ;b1 ,b2 ;−4x2 ,−4y 2 ]. 2 2 2 (ii) In (2.1), if we take i=1 and use the results (1.9)–(1.12), we get after a little simplification the following integral formula : Z ∞ 1 e−s sa−1 Ψ2 (b1 ;c1 ,c1 +1;xs,−xs)Ψ2 (b2 ;c2 ,c2 +1;ys,−ys)ds Γ(a) 0 1 1  a, a+ 12 : 12 b1 , 12 b1 + 21 ; 12 b2 , 12 b2 + 21 ; 2:2;2 2 2 −4x2 ,−4y 2  =F 0:3;3 1 1 1 1 1 1 − :c1 , 2 c1 + 2 , 2 c1 +1;c2 , 2 c2 + 2 , 2 c2 +1; +

ab1 x c1 (c1 +1)

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General integral formulas involving Humbert hypergeometric. . .

 1 a+ 21 , 12 a+1: 12 b1 + 12 , 12 b1 +1 ; 21 b2 , 21 b2 + 12 ; 2 2:2;2 −4x2 ,−4y 2  ×F 0:3;3 1 3 1 1 1 1 − :c1 +1, 2 c1 +1, 2 c1 + 2 ;c2 , 2 c2 + 2 , 2 c2 +1; ab2 y c2 (c2 +1) 1  a+ 21 , 12 a+1: 12 b1 , 12 b1 + 21 ; 12 b2 + 12 , 12 b2 +1 ; 2:2;2 2 −4x2 ,−4y 2  ×F 0:3;3 1 1 1 1 3 1 − :c1 , 2 c1 + 2 , 2 c1 +1;c2 +1, 2 c2 +1, 2 c2 + 2 ; +

+

a(a+1)b1 b2 xy c1 c2 (c1 +1)(c2 +1)

(3.5) 1  1 3 1 1 1 1 1 1 a+1, a+ : b + , b +1 ; b + , b +1 ; 1 1 2 2 2 2 2 2 2 2 2 2 2:2;2 2 −4x2 ,−4y 2 . ×F 0:3;3 1 1 3 1 1 3 − :c1 +1, 2 c1 +1, 2 c1 + 2 ;c2 +1, 2 c2 +1, 2 c2 + 2 ; Further, taking c1 =b1 −1, c2 =b2 −1 in (3.5) , we get Z ∞ 1 e−s sa−1 Ψ2 (b1 ;b1 −1,b1 ;xs,−xs)Ψ2 (b2 ;b2 −1,b2 ;ys,−ys)ds Γ(a) 0  =F4

1 1 1 a, a+ ;b1 −1,b2 −1;−4x2 ,−4y 2 2 2 2



  ax 1 1 1 2 2 + F4 a+ , a+1;b1 ,b2 −1;−4x ,−4y b1 −1 2 2 2   ay 1 1 1 2 2 + F4 a+ , a+1;b1 −1,b2 ;−4x ,−4y b2 −1 2 2 2

(3.6)

  a(a+1)xy 1 1 3 2 2 + F4 a+1, a+ ;b1 ,b2 ;−4x ,−4y . (b1 −1)(b2 −1) 2 2 2

(iii) In (2.2), if we take i=0 and use the results (1.9)–(1.12), we get after a little simplification the following integral formula : Z ∞ 1 e−s sa−1 Φ2 (b1 ,b1 ;c1 ;xs,−xs)Φ2 (b2 ,b2 ;c2 ;ys,−ys)ds Γ(a) 0 1 1  (3.7) a, a+ 12 : b1 ; b2 ; 2:1;1 2 2 x2 ,y 2 . =F 0:2;2 1 1 1 1 1 1 − : 2 c1 , 2 c1 + 2 ; 2 c2 , 2 c2 + 2 ;

Ahmed Ali Atash and Hussein Saleh Bellehaj

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Further, taking c1 =2b1 , c2 =2b2 in (3.7) and using the result [7] Φ2 (a,a;2a;x,y)=ey 1 F1 (a;2a;x,−y),

(3.8) we get 1 Γ(a)

Z

∞

e−s−xs−ys sa−1 1 F1 (b1 ;2b1 ;2xs)1 F1 (b2 ;2b2 ;2ys)ds

0

1 1 1 1 1 =F4 [ a, a+ ;b1 + ,b2 + ;x2 ,y 2 ]. 2 2 2 2 2 (iv) In (2.2), if we take i=1 and use the results (1.9)–(1.12), we get after a little simplification the following integral formula : Z ∞ 1 e−s sa−1 Φ2 (b1 −1,b1 ;c1 ;xs,−xs)Φ2 (b2 −1,b2 ;c2 ;ys,−ys)ds Γ(a) 0 1 1  1 a, a+ : b ; b ; 1 2 2 2:1;1 2 2 x2 ,y 2  =F 0:2;2 1 1 1 1 1 1 − : 2 c1 , 2 c1 + 2 ; 2 c2 , 2 c2 + 2 ; 1  a+ 12 , 21 a+1: b1 ; b2 ; 2 ax 2:1;1 x2 ,y 2  − F c1 0:2;2 − : 12 c1 + 21 , 21 c1 +1; 12 c2 , 21 c2 + 21 ; 1  a+ 12 , 21 a+1: b1 ; b2 ; ay 2:1;1 2 x2 ,y 2  − F c2 0:2;2 1 1 1 1 1 1 − : 2 c1 , 2 c1 + 2 ; 2 c2 + 2 , 2 c2 +1; (3.9)

a(a+1)xy c1 c2  1 a+1, 12 a+ 32 : b1 ; b2 ; 2 2:1;1 x2 ,y 2 . F 0:2;2 1 1 1 1 1 1 − : 2 c1 + 2 , 2 c1 +1; 2 c2 + 2 , 2 c2 +1; +

(3.10)

Further, taking c1 =2b1 −1, c2 =2b2 −1 in (3.10) , we get Z ∞ 1 e−s sa−1 Φ2 (b1 −1,b1 ;2b1 −1;xs,−xs) Γ(a) 0 ×Φ2 (b2 −1,b2 ;2b2 −1;ys,−ys)ds   1 1 1 1 1 2 2 =F4 a, a+ ;b1 − ,b2 − ;x ,y 2 2 2 2 2   ax 1 1 1 1 2 2 1 − F4 a+ , a+1;b1 + ,b2 − ;x ,y 2b1 −1 2 2 2 2 2

84

General integral formulas involving Humbert hypergeometric. . .

  ay 1 1 1 1 1 2 2 − F4 a+ , a+1;b1 − ,b2 + ;x ,y 2b2 −1 2 2 2 2 2

(3.11)

  1 a(a+1)xy 1 3 1 1 2 2 + F4 a+1, a+ ;b1 + ,b2 + ;x ,y . (2b1 −1)(2b2 −1) 2 2 2 2 2

The other special cases of the integrals (2.1) and (2.2) can also be obtained in the similar manner. References [1] W. N. Bailey, Generalized Hypergeometric Series, Cambridge University Press, Cambridge, 1935. [2] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi , Higher Transcendental Functions, Vol. I, McGraw-Hill, New York, Toronto and London, 1953. [3] H. Exton, Certain hypergeometric functions of four variables, Bull. Soc. Math. Gréce,(N.S.) 13 (1972), 104 – 113. [4] H. Exton, Multiple Hypergeometric Functions and Applications, John Wiley and Sons (Halsted Press), New York, 1976. [5] J. L. Lavoie, F. Grondin and A. K. Rathie, Generalizations of Whipple’s theorem on the sum of a 3 F2 , Journal of Computational and Applied Mathematics, 72 (1996), 293 – 300. [6] E. D. Rainville, Special Functions, The Macmillan Company, New York, 1960. [7] H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric series, Halsted Press , New York, 1985. [8] H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Halsted Press, New York, 1984.

Ahmed Ali Atash Department of Mathematics Faculty of Education-Shabwah Aden University Aden Yemen E-mail address: [email protected]

Hussein Saleh Bellehaj Department of Mathematics Faculty of Education-Shabwa Aden University Aden Yemen