Bulletin of the Korean Mathematical Society (대한수학회보)
- Volume 33 Issue 1
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- Pages.107-110
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- 1996
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- 1015-8634(pISSN)
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- 2234-3016(eISSN)
A reducible case of double hypergeometric series involving the riemann $zeta$-function
- Park, Junesang (Department of Mathematics, Dongguk University, Kyongju 780-714) ;
- H. M. Srivastava (Department of Mathematics, and Statistics, University, of Victoria, Victoria, British Columbia V8W 3P4)
- Published : 1996.02.01
Abstract
Usng the Pochhammer symbol $(\lambda)_n$ given by $$ (1.1) (\lambda)_n = {1, if n = 0 {\lambda(\lambda + 1) \cdots (\lambda + n - 1), if n \in N = {1, 2, 3, \ldots}, $$ we define a general double hypergeometric series by [3, pp.27] $$ (1.2) F_{q:s;\upsilon}^{p:r;u} [\alpha_1, \ldots, \alpha_p : \gamma_1, \ldots, \gamma_r; \lambda_1, \ldots, \lambda_u;_{x,y}][\beta_1, \ldots, \beta_q : \delta_1, \ldots, \delta_s; \mu_1, \ldots, \mu_v; ] = \sum_{l,m = 0}^{\infty} \frac {\prod_{j=1}^{q} (\beta_j)_{l+m} \prod_{j=1}^{s} (\delta_j)_l \prod_{j=1}^{v} (\mu_j)_m)}{\prod_{j=1}^{p} (\alpha_j)_{l+m} \prod_{j=1}^{r} (\gamma_j)_l \prod_{j=1}^{u} (\lambda_j)_m} \frac{l!}{x^l} \frac{m!}{y^m} $$ provided that the double series converges.