Very well-poised hypergeometric series and multiple integrals. (English. Russian original) Zbl 1229.33011

Zudilin, V. V.

Russ. Math. Surv. 57, No. 4, 824-826 (2002); translation from Usp. Mat. Nauk 57, No. 4, 177-178 (2002).

Summary: The purpose of this note is to establish a relationship (under certain conditions) between two objects: the very well-poised hypergeometric series \[ F_{k+2}(h_0;h_1,\dots,h_{k+1})=\sum_{\mu=0}^\infty(h_0+2\mu) \frac{\prod_{j=0}^{k+2}\Gamma(1+h_j+\mu)} {\Gamma(1+h_0-h_j+\mu)}(-1)^{(k+3)\mu} \] and the multiple integrals \[ J_k\left(\begin{matrix} h_1,h_2,\dots, h_{k+1} 1+h_0-h_3,\dots,1-h_0-h_{k+2}\end{matrix}\right)=\int\dots\int_{[0,1]^k}\frac{\prod_{j=1}^kx_j^{h_{j+1}-1} (1-x_j)^{h_0-h_{j+2}-h_{j+1}}}{(1-(1-(\cdots(1-x_k) x_{k-1})\cdots)x_2)x_1h_1}dx_1dx_2\cdots dx_k \] by the factor \[ \frac{\prod_{j=1}^{k+1}\Gamma(1+h_0-h_j-h_{j+1})} {\Gamma(h_1)\Gamma(h_{k+2})}. \]

MSC:

33C20	Generalized hypergeometric series, \({}_pF_q\)
33C60	Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)
11J82	Measures of irrationality and of transcendence
11J91	Transcendence theory of other special functions