Certain Generalized Appell Type Functions and Their Properties

Applied Mathematical Sciences, Vol. 9, 2015, no. 132, 6567 - 6581 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2015.59567

Certain Generalized Appell Type Functions and Their Properties Junesang Choi Department of Mathematics, Dongguk University Gyeongju 780-714, Republic of Korea Kottakkaran Sooppy Nisar Department of Mathematics, College of Arts and Science Prince Sattam bin Abdulaziz University Wadi Aldawaser, Riyadh region 11991, Saudi Arabia S. Jain Department of Mathematics Poornima College of Engineering Jaipur-302029, India Praveen Agarwal Department of Mathematics Anand International College of Engineering Jaipur-303012, India c 2015 Junesang Choi et al. This article is distributed under the Creative Copyright Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract Appell considered the product of two Gauss’s hypergeometric functions 2 F1 to devise four Appell’s functions F1 , F2 , F3 and F4 in two variables. Burchnall and Chaundy, and several others, systematically, presented a number of expansion and decomposition formulas for some double hypergeometric functions, for example, the Appell’s functions Fi , in series of simpler hypergeometric functions. Recently, Khan and

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Junesang Choi et al. Abukhammash introduced and investigated 10 Appell type generalized functions Mi (i = 1, . . . , 10) by considering the product of two 3 F2 functions. Here, motivated essentially by the above-mentioned works, we aim to introduce 18 Appell type generalized functions κi (i = 1, . . . , 18) by considering the product of two 4 F3 functions and, among other things, investigate their integral representations. We also present some decomposition formulas of κi (i = 1, · · · , 1 8) and certain relationships among the κi (i = 1, · · · , 1 8) by using symbolic operators.

Mathematics Subject Classification: Primary 42C05, Secondary 33C45 Keywords: Appell’s type functions; Hypergeometric series; Symbolic operators; Decomposition formulas; Integral representations

1

Introduction, Preliminaries and Definitions

The extensive development of the theories of hypergeometric functions of a single variable has led to a full-scale investigation of corresponding theories in two or more variables (see [2, 17, 18]; for recent developments, see also [3, 4, 8]). In 1880, Appell [1] considered the product of two Gauss’s hypergeometric functions to devise four Appell’s functions F1 , F2 , F3 and F4 in two variables. Later 1893, Lauricilla [12] further generalized the four Appell functions Fi (n) (n) (n) (n) (i = 1, 2, 3, 4) to give the functions FA , FB , FC and FD in n-variables. (2) (2) (1) (1) (1) (1) It is noted that FA = FB = FC = FD = 2 F1 , FA = F2 , FB = F3 , (2) (2) FC = F3 , and FD = F1 . Over seven decades ago, Burchnall and Chaundy [5, 6] and Chaundy [7] systematically presented a number of expansion and decomposition formulas for some double hypergeometric functions in series of simpler hypergeometric functions. Their method is based upon the following inverse pairs of symbolic operators: Γ(h)Γ(δ + δ 0 + h) ∇(h) = (1) Γ (δ + h)Γ (δ 0 + h) and ∆(h) = where

Γ (δ + h)Γ (δ 0 + h) , Γ(h)Γ(δ + δ 0 + h)

(2)

∂ ∂ and δ 0 ≡ y . ∂x ∂y For various multivariable hypergeometric functions including the Lauricella (n) (n) (n) (n) multivariable functions FA , FB , FC and FD , Hasanov and Srivastava [10, 11] presented a number of decomposition formulas in terms of such simpler hypergeometric functions as the Gauss and Appell functions. In the sequel to δ≡ x

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Certain generalized Appell type functions

the works of Hasanov and Srivastava [10, 11], using certain new inverse pairs of symbolic operators which were modified from (1) and (2), Choi and Hasanov [8] showed how some rather elementary techniques would lead easily to several decomposition formulas associated with Humbert’s hypergeometric functions Φ1 , Φ2 , Φ3 , Ψ1 , Ψ2 , Ξ1 and Ξ2 . Burchnall and Chaundy [5, 6] suggested a possible extension of their results to functions of higher order (with more parameters) for two variable as follows:   0 0 0 a; b1 , b2 , . . . , bp ; b1 , b2 , . . . , bp ; (2) x, y 0 0 0 p+1 Fp c1 , c2 , . . . , cp ; c1 , c2 , . . . , cp ; 0 0 0 ∞ X (a)m+n (b1 )m · · · (bp )m (b1 )n (b2 )n · · · (bp )n xm y n = (3) 0 0 0 m!n!(c1 )m (c2 )m · · · (cp )m (c1 )n (c2 )n · · · (cp )n m, n=0     0 0 0 a, b1 , b2 , . . . , bp ; a, b1 , b2 , . . . , bp ; y = ∇(a) p+1 Fp x p+1 Fp 0 0 0 c1 , c2 , . . . , cp ; c1 , c2 , . . . , cp ;   0 0 0 a; b1 , b2 , . . . , bp ; b1 , b2 , . . . , bp ; (2) x, y 0 0 0 p+1 Fp c1 , c2 , . . . , cp ; c1 , c2 , . . . , cp ; 0 0 0 ∞ X (a)r (b1 )r · · · (bp )r (b1 )r (b2 )r · · · (bp )r = xr y r 0 0 0 ) r!(c ) (c ) · · · (c ) (c ) (c ) · · · (c 1 r 2 r p r 1 r 2 r p r r=0 (4)   a + r, b1 + r, b2 + r, . . . , bp + r; × p+1 Fp x c1 + r, c2 + r, . . . , cp + r;   0 0 0 a + r, b1 + r, b2 + r, . . . , bp + r; × p+1 Fp y . 0 0 0 c1 , c2 , . . . , cp ; Recently, Khan and Abukhammash [9] introduced 10 Appell type generalized functions Mi (i = 1, . . . , 10) by considering the product of two 3 F2 functions. Here, in this paper, we consider the product of two 4 F3 hypergeometric functions, i.e., 0 0 0 0 0 0 0 4 F3 (a, b, c, d; e, f, g; x) 4 F3 (a , b , c , d ; e , f , g ; y) ∞ X (a)m (a0 )n (b)m (b0 )n (c)m (c0 )n (d)m (d0 )n = (e)m (e0 )n (f )m (f 0 )n (g)m (g 0 )n m, n=0

xm y n . m! n!

(5)

This double series, in itself, yields nothing new, but by replacing one or more of the seven pairs of products (a)m (a0 )n , (d)m (d0 )n ,

(b)m (b0 )n ,

(e)m (e0 )n ,

(c)m (c0 )n ,

(f )m (f 0 )n ,

(g)m (g 0 )n

by the corresponding expressions (a)m+n , (b)m+n , (c)m+n , (d)m+n , (e)m+n , (f )m+n ,

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we are led to nineteen distinct possibilities of getting new double series. One such possibility, however, gives us the double series ∞ X (a)m+n (b)m+n (c)m+n (d)m+n xm y n , (e)m+n (f )m+n (g)m+n m! n! m, n=0

which, upon using the well-known (easily-derivable) identity (see, e.g., [15]): ∞ X ∞ X

∞ X xm y n (x + y)N f (m + n) = f (N ) , m! n! N =0 N! m=0 n=0

(6)

is simply the hypergeometric series 4 F3 (a, b, c, d; e, f, g; x + y). The remaining possibilities lead to the following eighteen generalized Appell type functions of two variables: κ1 (a, a0 , b, b0 , c, c0 , d, d0 ; e, f, f 0 , g, g 0 ; x, y) ∞ X ∞ X (a)m (a0 )n (b)m (b0 )n (c)m (c0 )n (d)m (d0 )n xm y n := (e)m+n (f )m (f 0 )n (g)m (g 0 )n m! n! m=0 n=0

(7)

(max{|x|, |y|} < 1) ; κ2 (a, a0 , b, b0 , c, c0 , d, d0 ; e, f, g, g 0 ; x, y) ∞ X ∞ X (a)m (a0 )n (b)m (b0 )n (c)m (c0 )n (d)m (d0 )n xm y n := (e)m+n (f )m+n (g)m (g 0 )n m! n! m=0 n=0

(8)

(max{|x|, |y|} < 1) ; κ3 (a, a0 , b, b0 , c, c0 , d, d0 ; e, f, g; x, y) ∞ X ∞ X (a)m (a0 )n (b)m (b0 )n (c)m (c0 )n (d)m (d0 )n xm y n := (e)m+n (f )m+n (g)m+n m! n! m=0 n=0

(9)

(max{|x|, |y|} < 1) ; κ4 (a, b, b0 , c, c0 , d, d0 ; e, e0 , f, f 0 , g, g 0 ; x, y) ∞ X ∞ X (a)m+n (b)m (b0 )n (c)m (c0 )n (d)m (d0 )n xm y n := (e)m (e0 )n (f )m (f 0 )n (g)m (g 0 )n m! n! m=0 n=0

(10)

(|x| + |y| < 1) ; κ5 (a, b, b0 , c, c0 , d, d0 ; e, f, f 0 , g, g 0 ; x, y) ∞ X ∞ X (a)m+n (b)m (b0 )n (c)m (c0 )n (d)m (d0 )n xm y n := (e)m+n (f )m (f 0 )n (g)m (g 0 )n m! n! m=0 n=0 (max{|x|, |y|} < 1) ;

(11)

Certain generalized Appell type functions

κ6 (a, b, b0 , c, c0 , d, d0 ; e, f, g, g 0 ; x, y) ∞ X ∞ X (a)m+n (b)m (b0 )n (c)m (c0 )n (d)m (d0 )n xm y n := (e)m+n (f )m+n (g)m (g 0 )n m! n! m=0 n=0

6571

(12)

(max{|x|, |y|} < 1) ; κ7 (a, b, b0 , c, c0 , d, d0 ; e, f, g; x, y) ∞ X ∞ X (a)m+n (b)m (b0 )n (c)m (c0 )n (d)m (d0 )n xm y n := (e)m+n (f )m+n (g)m+n m! n! m=0 n=0

(13)

(max{|x|, |y|} < 1) ; κ8 (a, b, , c, c0 , d, d0 ; e, e0 , f, f 0 , g, g 0 ; x, y) ∞ X ∞ X (a)m+n (b)m+n (c)m (c0 )n (d)m (d0 )n xm y n := (e)m (e0 )n (f )m (f 0 )n (g)m (g 0 )n m! n! m=0 n=0 p  p |x| + |y| < 1 ; κ9 (a, b, c, c0 , d, d0 ; e, f, f 0 , g, g 0 ; x, y) ∞ X ∞ X (a)m+n (b)m+n (c)m (c0 )n (d)m (d0 )n xm y n := (e)m+n (f )m (f 0 )n (g)m (g 0 )n m! n! m=0 n=0

(14)

(15)

(|x| + |y| < 1) ; κ10 (a, b, c, c0 , d, d0 ; e, f, g, g 0 ; x, y) ∞ X ∞ X (a)m+n (b)m+n (c)m (c0 )n (d)m (d0 )n xm y n := (e)m+n (f )m+n (g)m (g 0 )n m! n! m=0 n=0

(16)

(max{|x|, |y|} < 1) ; κ11 (a, b, c, c0 , d, d0 ; e, f, g; x, y) ∞ X ∞ X (a)m+n (b)m+n (c)m (c0 )n (d)m (d0 )n xm y n := (e)m+n (f )m+n (g)m+n m! n! m=0 n=0

(17)

(max{|x|, |y|} < 1) ; κ12 (a, b, c, d, d0 ; e, e0 , f, f 0 , g, g 0 ; x, y) ∞ X ∞ X (a)m+n (b)m+n (c)m+n (d)m (d0 )n xm y n := (e)m (e0 )n (f )m (f 0 )n (g)m (g 0 )n m! n! m=0 n=0 p  p 3 |x| + 3 |y| < 1 ;

(18)

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κ13 (a, b, c, d, d0 ; e, f, f 0 , g, g 0 ; x, y) ∞ X ∞ X (a)m+n (b)m+n (c)m+n (d)m (d0 )n xm y n := (e)m+n (f )m (f 0 )n (g)m (g 0 )n m! n! m=0 n=0 p  p |x| + |y| < 1 ; κ14 (a, b, c, d, d0 ; e, f, g, g 0 ; x, y) ∞ X ∞ X (a)m+n (b)m+n (c)m+n (d)m (d0 )n xm y n := (e)m+n (f )m+n (g)m (g 0 )n m! n! m=0 n=0

(19)

(20)

(|x| + |y| < 1) ; κ15 (a, b, c, d, d0 ; e, f, g; x, y) ∞ X ∞ X (a)m+n (b)m+n (c)m+n (d)m (d0 )n xm y n := (e)m+n (f )m+n (g)m+n m! n! m=0 n=0

(21)

(max{|x|, |y|} < 1) ; κ16 (a, b, c, d; e, e0 , f, f 0 , g, g 0 ; x, y) ∞ X ∞ X (a)m+n (b)m+n (c)m+n (d)m+n xm y n := (e)m (e0 )n (f )m (f 0 )n (g)m (g 0 )n m! n! m=0 n=0 p  p 4 |x| + 4 |y| < 1 ;

(22)

κ17 (a, b, , c, d; e, f, f 0 , g, g 0 ; x, y) ∞ X ∞ X (a)m+n (b)m+n (c)m+n (d)m+n xm y n := (e)m+n (f )m (f 0 )n (g)m (g 0 )n m! n! m=0 n=0 p  p 3 |x| + 3 |y| < 1 ;

(23)

κ18 (a, b, , c, d; e, f, g, g 0 ; x, y) ∞ ∞ X X (a)m+n (b)m+n (c)m+n (d)m+n xm y n := (e)m+n (f )m+n (g)m (g 0 )n m! n! m=0 n=0 p  p |x| + |y| < 1 ,

(24)

where we exclude the exceptional parameter values, i.e., those values for which the series in question becomes terminating, meaningless, or a finite sum of hypergeometric series of lower dimension.

Certain generalized Appell type functions

2

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Symbolic Form

Here we follow the method given by Burchnall and Chaundy [5, 6], Chaundy [7], and Choi and Hasanov [8] to use the symbolic operators (1) and (2) to obtain the following decomposition formulas of our newly defined functions κi (i = 1, · · · , 1 8) asserted by Theorem 2.1. We also give certain interesting relationships among the κi (i = 1, · · · , 1 8) by using the symbolic operators (1) and (2) given in Theorem 2.2. Theorem 2.1. Let ∇(h) and ∆(h) be given in (1) and (2), respectively. Each of the following formulas holds true: κ1 (a, a0 , b, b0 , c, c0 , d, d0 ; e, f, f 0 , g, g 0 ; x, y) = ∆(e) 4 F3 (a, b, c, d; e, f, g; x) 4 F3 (a0 , b0 , c0 , d0 ; e, f 0 , g 0 ; y);

(25)

κ2 (a, a0 , b, b0 , c, c0 , d, d0 ; e, f, g, g 0 ; x, y) = ∆(e)∆(f ) 4 F3 (a, b, c, d; e, f, g; x) 4 F3 (a0 , b0 , c0 , d0 ; e, f, g 0 ; y);

(26)

κ3 (a, a0 , b, b0 , c, c0 , d, d0 ; e, f, g; x, y) = ∆(e)∆(f )∆(g) 4 F3 (a, b, c, d; e, f, g; x) 4 F3 (a0 , b0 , c0 , d0 ; e, f, g; y);

(27)

κ4 (a, b, b0 , c, c0 , d, d0 ; e, e0 f, f 0 , g, g 0 ; x, y) = ∇(a) 4 F3 (a, b, c, d; e, f, g; x) 4 F3 (a, b0 , c0 , d0 ; e0 , f 0 , g 0 ; y);

(28)

κ5 (a, b, b0 , c, c0 , d, d0 ; e, f, f 0 , g, g 0 ; x, y); ∇(a)∆(e) 4 F3 (a, b, c, d; e, f, g; x) 4 F3 (a, b0 , c0 , d0 ; e, f 0 , g 0 ; y);

(29)

κ6 (a, b, b0 , c, c0 , d, d0 ; e, f, g, g 0 ; x, y) = ∇(a)∆(e)∆(f ) 4 F3 (a, b, c, d; e, f, g; x) 4 F3 (a, b0 , c0 , d0 ; e, f, g 0 ; y);

(30)

κ7 (a, b, b0 , c, c0 , d, d0 ; e, f, g; x, y) = ∇(a)∆(e)∆(f )∆(g) 4 F3 (a, b, c, d; e, f, g; x) 4 F3 (a, b0 , c0 , d0 ; e, f, g; y); (31) κ8 (a, b, c, c0 , d, d0 ; e, e0 , f, f 0 , g, g 0 ; x, y) = ∇(a)∇(b) 4 F3 (a, b, c, d; e, f, g; x) 4 F3 (a, b, c0 , d0 ; e0 , f 0 , g 0 ; y);

(32)

κ9 (a, b, c, c0 , d, d0 ; e, f, f 0 , g, g 0 ; x, y) = ∇(a)∇(b)∆(e) 4 F3 (a, b, c, d; e, f, g; x) 4 F3 (a, b, c0 , d0 ; e, f 0 , g 0 ; y);

(33)

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κ10 (a, b, c, c0 , d, d0 ; e, f, g, g 0 ; x, y); = ∇(a)∇(b)∆(e)∆(f ) 4 F3 (a, b, c, d; e, f, g; x) 4 F3 (a, b, c0 , d0 ; e, f, g 0 ; y); (34) κ11 (a, b, c, c0 , d, d0 ; e, f, g; x, y); (35) = ∇(a)∇(b)∆(e)∆(f ) 4 F3 (a, b, c, d; e, f, g; x) 4 F3 (a, b, c0 , d0 ; e, f, g; y); κ12 (a, b, c, d, d0 ; e, e0 , f, f 0 , g, g 0 ; x, y) = ∇(a)∇(b)∇(c) 4 F3 (a, b, c, d; e, f, g; x) 4 F3 (a, b, c, d0 ; e0 , f 0 , g 0 ; y);

(36)

κ13 (a, b, c, d, d0 ; e, f, f 0 , g, g 0 ; x, y) (37) = ∇(a)∇(b)∇(c)∆(e) 4 F3 (a, b, c, d; e, f, g; x) 4 F3 (a, b, c, d0 ; e, f 0 , g 0 ; y); κ14 (a, b, c, d, d0 ; e, f, g, g 0 ; x, y) = ∇(a)∇(b)∇(c)∆(e)∆(f ) ×4 F3 (a, b, c, d; e, f, g; x) 4 F3 (a, b, c, d0 ; e, f, g 0 ; y);

(38)

κ15 (a, b, c, d, d0 ; e, f, g; x, y) = ∇(a)∇(b)∇(c)∆(e)∆(f )∆(g) ×4 F3 (a, b, c, d; e, f, g; x) 4 F3 (a, b, c, d0 ; e, f, g; y);

(39)

κ16 (a, b, c, d ; e, e0 , f, f 0 , g, g 0 ; x, y) = ∇(a)∇(b)∇(c)∇(d) ×4 F3 (a, b, c, d; e, f, g; x) 4 F3 (a, b, c, d; e0 , f 0 , g 0 ; y);

(40)

κ17 (a, b, c, d ; e, f, f 0 , g, g 0 ; x, y) = ∇(a)∇(b)∇(c)∇(d)∆(e) × 4 F3 (a, b, c, d; e, f, g; x) 4 F3 (a, b, c, d; e, f 0 , g 0 ; y);

(41)

κ18 (a, b, c, d ; e, f, g, g 0 ; x, y) ∇(a)∇(b)∇(c)∇(d)∆(e)∆(f ) × 4 F3 (a, b, c, d; e, f, g; x) 4 F3 (a, b, c, d; e, f, g 0 ; y).

(42)

Proof. The results presented here can be easily derived by just following the method in Burchnall and Chaundy [5, 6], (see also [7], [8]). So the details of proof are omitted. Theorem 2.2. Let ∇(h) and ∆(h) be given in (1) and (2), respectively. Each of the following relationships holds true: κ1 (a, a, b, b0 , c, c0 , d, d0 ; e, f, f 0 , g, g 0 ; x, y) = ∆(a) κ5 (a, b, b0 , c, c0 , d, d0 ; e, f, f 0 , g, g 0 ; x, y) ;

(43)

κ5 (a, b, b0 , c, c0 , d, d0 ; e, f, f 0 , g, g 0 ; x, y) = ∇(a) κ1 (a, a, b, b0 , c, c0 , d, d0 ; e, f, f 0 , g, g 0 ; x, y) ;

(44)

Certain generalized Appell type functions

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κ2 (a, a, b, b0 , c, c0 , d, d0 ; e, f, g, g 0 ; x, y) = ∆(a) κ6 (a, b, b0 , c, c0 , d, d0 ; e, f, g, g 0 ; x, y) ;

(45)

κ6 (a, b, b0 , c, c0 , d, d0 ; e, f, g, g 0 ; x, y) = ∇(a) κ2 (a, a, b, b0 , c, c0 , d, d0 ; e, f, g, g 0 ; x, y) ;

(46)

κ3 (a, a0 , b, b0 , c, c0 , d, d0 ; e, f, g; x, y) = ∆(g) κ2 (a, a0 , b, b0 , c, c0 , d, d0 ; e, f, g, g; x, y) ;

(47)

κ2 (a, a0 , b, b0 , c, c0 , d, d0 ; e, f, g, g; x, y) = ∇(g) κ3 (a, a0 , b, b0 , c, c0 , d, d0 ; e, f, g; x, y) ;

(48)

κ4 (a, b, b, c, c0 , d, d0 ; e, e0 , f, f 0 , g, g 0 ; x, y) = ∆(b) κ8 (a, b, c, c0 , d, d0 ; e, e0 , f, f 0 , g, g 0 ; x, y) ;

(49)

κ8 ( a, b, c, c0 , d, d0 ; e, e0 , f, f 0 , g, g 0 ; x, y) = ∇(b) κ4 (a, b, b, c, c0 , d, d0 ; e, e0 , f, f 0 , g, g 0 ; x, y) ;

(50)

κ7 (a, b, b0 , c, c0 , d, d0 ; e, f, g; x, y) = ∇(a) κ3 (a, a, b, b0 , c, c0 , d, d0 ; e, f, g; x, y) ;

(51)

κ3 (a, a, b, b0 , c, c0 , d, d0 ; e, f, g; x, y) = ∆(a) κ7 (a, b, b0 , c, c0 , d, d0 ; e, f, g; x, y) ;

(52)

κ5 (a, b, b, c, c0 , d, d0 ; e, f, f 0 , g, g 0 ; x, y) = ∆(b) κ9 (a, b, c, c0 , d, d0 ; e, f, f 0 , g, g 0 ; x, y) ;

(53)

κ9 (a, b, c, c0 , d, d0 ; e, f, f 0 , g, g 0 ; x, y) = ∇(b) κ5 (a, b, b, c, c0 , d, d0 ; e, f, f 0 , g, g 0 ; x, y) ;

(54)

κ10 (a, b, c, c0 , d, d0 ; e, f, g, g 0 ; x, y) = ∇(b) κ6 (a, b, b, c, c0 , d, d0 ; e, f, g, g 0 ; x, y) ;

(55)

κ6 (a, b, b, c, c0 , d, d0 ; e, f, g, g 0 ; x, y) = ∆(b) κ10 (a, b, c, c0 , d, d0 ; e, f, g, g 0 ; x, y) ;

(56)

κ11 (a, b, c, c0 , d, d0 ; e, f, g; x, y) = ∇(b) κ7 (a, b, b, c, c0 , d, d0 ; e, f, g; x, y) ;

(57)

κ7 (a, b, b, c, c0 , d, d0 ; e, f, g; x, y) = ∆(b) κ11 (a, b, c, c0 , d, d0 ; e, f, g; x, y) ;

(58)

κ13 ( a, b, c, d, d0 ; e, f, f 0 , g, g 0 ; x, y) = ∇(c) κ9 (a, b, c, c, d, d0 ; e, f, f 0 , g, g 0 ; x, y) ;

(59)

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κ9 (a, b, c, c, d, d0 ; e, f, f 0 , g, g 0 ; x, y) = ∆(c) κ13 ( a, b, c, d, d0 ; e, f, f 0 , g, g 0 ; x, y) ;

(60)

κ13 (a, b, c, d, d0 ; e, f, f 0 , g, g 0 ; x, y) = ∆(e) κ12 (a, b, c, d, d0 ; e, e, f, f 0 , g, g 0 ; x, y) ;

(61)

κ12 (a, b, c, d, d0 ; e, e, f, f 0 , g, g 0 ; x, y) = ∇(e) κ13 ( a, b, c, d, d0 ; e, f, f 0 , g, g 0 ; x, y) ;

(62)

κ14 (a, b, c, , d, d0 ; e, f, g, g 0 ; x, y) = ∆(f ) κ13 (a, b, c, d, d0 ; e, f, f, g, g 0 ; x, y) ;

(63)

κ13 (a, b, c, d, d0 ; e, f, f, g, g 0 ; x, y) = ∇(f ) κ14 (a, b, c, d, d0 ; e, f, g, g 0 ; x, y) ;

(64)

κ15 (a, b, c, d, d0 ; e, f, g; x, y) = ∆(g) κ14 (a, b, c, d, d0 ; e, f, g, g; x, y) ;

(65)

κ14 (a, b, c, d, d0 ; e, f, g, g; x, y) = ∇(g) κ15 (a, b, c, d, d0 ; e, f, g; x, y) ;

(66)

κ16 ( a, b, c, d; e, e, f, f 0 , g, g; x, y) = ∇(e) κ17 ( a, b, c, d; e, f, f 0 , g, g 0 ; x, y) ;

(67)

κ17 ( a, b, c, d; e, f, f 0 , g, g 0 ; x, y) = ∆(e)κ16 ( a, b, c, d; e, e, f, f 0 , g, g 0 ; x, y) ;

(68)

κ18 ( a, b, c, d; e, f, g, g 0 ; x, y) = ∆(f ) κ17 ( a, b, c, d; e, f, f, g, g 0 ; x, y) ;

(69)

κ17 ( a, b, c, d; e, f, f, g, g; x, y) = ∇(f ) κ18 ( a, b, c, d; e, f, g, g 0 ; x, y) ;

(70)

κ15 ( a, b, c, d, d0 ; e, f, g; x, y) = ∆(f )∆(g) κ13 ( a, b, c, d, d0 ; e, f, f, g, g; x, y) ;

(71)

κ13 ( a, b, c, d, d0 ; e, f, f, g, g; x, y) = ∇(f )∇(g) κ15 ( a, b, c, d, d0 ; e, f, g; x, y) ;

(72)

κ18 ( a, b, c, d; e, f, g, g 0 ; x, y) = ∆(e)∆(f ) κ16 ( a, b, c, d; e, e, f, f, g, g 0 ; x, y) ;

(73)

κ16 ( a, b, c, d; e, e, f, f, g, g 0 ; x, y) = ∇(f )∇(e) κ18 ( a, b, c, d; e, f, g, g 0 ; x, y) .

(74)

6577

Certain generalized Appell type functions

Proof. The results presented here can be easily derived by just following the method in Burchnall and Chaundy [5, 6], (see also [7], [8]). So the details of proof are omitted.

3

Integral Representations

Here we present certain integral representations for the functions κi . The following well-known integral Eulerian formulas are required: Z 1 Γ(α)Γ(β) uα−1 (1 − u)β−1 du = ( 0, 0) (75) Γ(α + β) 0 and Z Z Γ(α)Γ(β)Γ(γ) uα−1 v β−1 (1 − u − v)γ−1 dudv = Γ(α + β + γ) (76) (u ≥ 0, v ≥ 0, u + v ≤ 1, min {