Commun. Korean Math. Soc. 28 (2013), No. 2, pp. 297–301 http://dx.doi.org/10.4134/CKMS.2013.28.2.297
A REDUCIBILITY OF SRIVASTAVA’S TRIPLE HYPERGEOMETRIC SERIES F (3) [x, y, z] Junesang Choi, Xiaoxia Wang, and Arjun K. Rathie Abstract. When certain general single or multiple hypergeometric functions were introduced, their reduction formulas have naturally been investigated. Here, in this paper, we aim at presenting a very interesting reduction formula for the Srivastava’s triple hypergeometric function F (3) [x, y, z] by applying the so-called Beta integral method to the Henrici’s triple product formula for hypergeometric series.
1. Introduction In the usual notation, let C denote the set of complex numbers. For αj ∈ C (j = 1, . . . , p) and βj ∈ C \ Z− 0
Z− 0 := Z ∪ {0} = {0, −1, −2, . . .} ,
the generalized hypergeometric function p Fq with p numerator parameters α1 , . . . , αp and q denominator parameters β1 , . . . , βq is defined by (see, for example, [6, Chapter 4]; see also [9, pp. 71–72]):
(1.1)
p Fq
α1 , . . . , αp ; β1 , . . . , βq ;
z =
∞ X
n=0
p Y
j=1 q Y
(αj )n (βj )n
zn n!
j=1
= p Fq (α1 , . . . , αp ; β1 , . . . , βq ; z)
p, q ∈ N0 := N ∪ {0} = {0, 1, 2, . . .}; p ≦ q + 1; p ≦ q and |z| < ∞; p = q + 1 and |z| < 1; p = q + 1, |z| = 1 and ℜ(ω) > 0 ,
Received April 20, 2012; Revised December 28, 2012. 2010 Mathematics Subject Classification. Primary 33C20, 33B20; Secondary 33C90, 33C05. Key words and phrases. generalized hypergeometric function p Fq , Gamma function, Pochhammer symbol, Beta integral, Srivastava’s triple hypergeometric series F (3) [x, y, z], Henrici’s formula. c
2013 The Korean Mathematical Society
297
298
JUNESANG CHOI, XIAOXIA WANG, AND ARJUN K. RATHIE
where (1.2) ω :=
q X j=1
βj −
p X j=1
αj αj ∈ C (j = 1, . . . , p); βj ∈ C \ Z− 0 (j = 1, . . . , q)
and (λ)n is the Pochhammer symbol defined (for λ ∈ C), in terms of the familiar Gamma function Γ, by ( 1 (n = 0) Γ(λ + n) (1.3) (λ)n := = Γ(λ) λ(λ + 1) · · · (λ + n − 1) (n ∈ N). A unification of Lauricella’s 14 triple hypergeometric series F1 , . . . , F14 (cf. [10, pp. 41–43]) and the additional Srivastava’s triple hypergeometric series HA , HB , HC was introduced by Srivastava [7] (cf. [10, p. 43]) who defined a general triple hypergeometric series F (3) [x, y, z] (cf. [7, p. 428]; see also [10, pp. 44–45]): (a) :: (b); (b′ ); (b′′ ) : (c); (c′ ); (c′′ ); F (3) [x, y, z] ≡ F (3) x, y, z (e) :: (g); (g ′ ); (g ′′ ) : (h); (h′ ); (h′′ ); (1.4) ∞ X xm y n z p , = Λ(m, n, p) m! n! p! m,n,p=0 where, for convenience,
Λ(m, n, p) =
(1.5)
A Y
B Y
(aj )m+n+p
(bj )m+n
j=1
j=1
j=1
E Y
G Y
G Y
(ej )m+n+p
·
C Y
(cj )m
C′ Y
j=1
j=1
H Y
H′ Y
j=1
(hj )m
j=1
(b′j )n+p
(gj )m+n
(c′j )n
(gj′′ )p+m
j=1
,
′′
H Y
G Y
(c′′j )p
j=1
(h′j )n
(b′′j )p+m
′′
(gj′ )n+p
j=1 C ′′ Y
B Y
j=1
′
j=1
j=1
′′
′
B Y
(h′′j )p
j=1
and, (a) abbreviates the array of A parameters a1 , . . . , aA , with similar interpretations for (b), (b′ ), (b′′ ), and so on. When general hypergeometric functions in one, two and more variables were given, their reduction formulas have naturally been investigated. For various interesting reduction formulas, see, for example, [10, pp. 32, 34, 38, and so on], and see also [1], [2] and [8]. Here, we would like to mention an interesting reduction formula for the Srivastava’s triple hypergeometric series F (3) [−x/(1− z)(1 − x), −y/(1 − x)(1 − y), −z/(1 − y)(1 − z)] (see [10, p. 272, Equation (17);
A REDUCIBILITY OF SRIVASTAVA’S TRIPLE HYPERGEOMETRIC SERIES
p. 272, Equation (121)]): (1.6) :: β; β ′ ; α : ; ; ; (3) F − ′ :: ; ;
: α; β; β ;
299
y z x ,− ,− (1 − z)(1 − x) (1 − x)(1 − y) (1 − y)(1 − z) ′
= (1 + xyz)−1 (1 − x)β (1 − y)β (1 − z)α .
Here, in this paper, we aim at presenting a very interesting reduction formula for the Srivastava’s triple hypergeometric function F (3) [x, y, z] by applying the so-called Beta integral method (see, for example, [5]) to the Henrici’s triple product formula for hypergeometric series [3]: ; ; ; 2 x ωx ω x F F F 0 1 0 1 0 1 6c; 6c; 6c; " 3 # (1.7) 4x 3c − 14 , 3c + 14 ; , = 2 F7 1 2 1 1 6c, 2c, 2c + 3 , 2c + 3 , 4c − 3 , 4c, 4c + 3 ; 9 where ω = exp( 2πi 3 ). 2. Main theorem We begin by stating our main theorem. Theorem. The following interesting reduction formula for the Srivastava’s triple hypergeometric function F (3) [x, y, z] holds true. (2.1) ; ; : ; ; ; e :: 1, ω, ω 2 F (3) d :: ; ; : 6c; 6c; 6c; " 3 # 4 3c − 41 , 3c + 14 , 3e , 3e + 13 , 3e + 32 ; = 5 F10 , 6c, 2c, 2c + 13 , 2c + 32 , 4c − 13 , 4c, 4c + 31 , d3 , d3 + 31 , d3 + 32 ; 9 where ω = exp( 2πi 3 ). Proof. Multiply the left-hand side of (1.7) by xe−1 (1 − x)d−e−1 , integrate the resulting equation with respect to x between 0 to 1, express the involved three 0 F1 as series, change the order of summation and integration which is easily seen to be justified due to the uniform convergence of the involved series, finally use the following well-known relationship between the Beta function B(α, β) and the Gamma function Γ (see, for example, [9, Section 1.1]): Z 1 tα−1 (1 − t)β−1 dt (ℜ(α) > 0; ℜ(β) > 0) 0 (2.2) B(α, β) = Γ(α) Γ(β) (α, β ∈ C \ Z− 0 ), Γ(α + β)
300
JUNESANG CHOI, XIAOXIA WANG, AND ARJUN K. RATHIE
and, after a little simplification, we get Γ(e)Γ(d − e) Γ(d)
∞ X
m,n,p=0
(e)m+n+p ω n ω 2p . (d)m+n+p (6c)m (6c)n (6c)p m! n! p!
In view of the Srivastava’s function (1.4), we see that the last triple series is interpreted as follows: Γ(e)Γ(d − e) (3) e :: ; ; : ; ; ; 2 F 1, ω, ω . (2.3) d :: ; ; : 6c; 6c; 6c; Γ(d) On the other hand, multiply the right-hand side of (1.7) by xe−1 (1 − x)d−e−1 and proceed as above. Then, applying the multiplication formula for the Gamma function (in case of m = 3): m Y 1 j−1 1 Γ(mz) = (2π) 2 (1−m) mmz− 2 Γ z+ m j=1 (2.4) 1 2 z 6= 0, − , − , · · · ; m ∈ N m m to the last resulting series, we have ∞ 3c − 14 n 3c + 41 n e3 n Γ(e)Γ(d − e) X Γ(d) n! (6c)n (2c)n 2c + 31 n 2c + 23 n 4c − 31 n (4c)n n=0 e 2 4 3n 1 e 3 + 3 n 3 + 3 n 9 . · 4c + 13 n d3 n d3 + 31 n d3 + 32 n
If we express this last series in terms of the generalized hypergeometric series (1.1), we have (2.5) " # Γ(e)Γ(d − e) 5 F10 Γ(d)
6c, 2c, 2c +
1 3,
2c +
2 3,
3c − 14 , 3c + 41 , 3e , e3 + 31 , 4c − 13 , 4c, 4c + 31 , d3 , d3 + 31 ,
e 3 d 3
+ 32 ; + 32 ;
Finally, equating the two expressions (2.3) and (2.5) proves (2.1).
3 4 . 9
Remark. Karlsson and Srivastava [4] generalized (1.7) by utilizing certain known transformations of hypergeometric functions and deduced (1.7) as a special case. Krattenthaler and Rao [5] made a systematic use of the so-called Beta integral method, a method of deriving new hypergeometric identities from old ones by mainly using the Beta integral in (2.2) based on the Mathematica Package HYP, to illustrate several interesting identities for the hypergeometric series and Kamp´e de F´eriet series of unit arguments. Acknowledgements. The present investigation was supported, in part, by Shanghai Leading Academic Discipline Project S30104.
A REDUCIBILITY OF SRIVASTAVA’S TRIPLE HYPERGEOMETRIC SERIES
301
References [1] R. G. Buschman and H. M. Srivastava, Series identities and reducibility of Kamp´ e de F´ eriet functions, Math. Proc. Cambridge Philos. Soc. 91 (1982), no. 3, 435–440. [2] C. C. Grosjean and H. M. Srivastava, Some transformation and reduction formulas for hypergeometric series in several variables, J. Comput. Appl. Math. 37 (1991), no. 1-3, 287–299. [3] P. Henrici, A triple product theorem for hypergeometric series, SIAM J. Math. Anal. 18 (1987), no. 6, 1513–1518. [4] P. W. Karlsson and H. M. Srivastava, A note on Henrici’s triple product theorem, Proc. Amer. Math. Soc. 110 (1990), no. 1, 85–88. [5] C. Krattenthaler and K. S. Rao, Automatic generation of hypergeometric identities by the beta integral method, J. Comput. Appl. Math. 160 (2003), no. 1-2, 159–173. [6] E. D. Rainville, Special Functions, Macmillan Company, New York, 1960; Reprinted by Chelsea Publishing Company, Bronx, New York, 1971. [7] H. M. Srivastava, Generalized Neumann expansions involving hypergeometric functions, Proc. Cambridge Philos. Soc. 63 (1967), 425–429. [8] , On the reducibility of Appell’s function F4 , Canad. Math. Bull. 16 (1973), 295–298. [9] H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London and New York, 2012. [10] H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1985. Junesang Choi Department of Mathematics Dongguk University Gyeongju 780-714, Korea E-mail address:
[email protected] Xiaoxia Wang Department of Mathematics Shanghai University Shanghai 200444, P. R. China E-mail address:
[email protected] Arjun K. Rathie Department of Mathematics School of Mathematical & Physical Sciences Central University of Kerala Riverside Transit Campus Padennakad P.O. Nileshwar, Kasaragod-671 328, India E-mail address:
[email protected]
