New Analytic Representations of the Hypergeometric Functions $${}_{p+1}$$F$$_p$$

Abstract

The power series expansions of the hypergeometric functions \({}_{p+1}F_p(a,b_1,\ldots ,b_p;c_1,\ldots ,c_p;z)\) converge either inside the unit disk \(\vert z\vert <1\) or outside this disk \(\vert z\vert >1\). Nørlund’s expansion in powers of \(z/(z-1)\) converges in the half-plane \(\mathfrak {R}(z)<1/2\). For arbitrary \(z_0 \in \mathbb {C}\), Bühring’s expansion in inverse powers of \(z-z_0\) converges outside the disk \(\vert z-z_0\vert =\) max\(\lbrace \vert z_0\vert ,\vert z_0-1\vert \rbrace \). None of them converge on the whole indented closed unit disk \(\vert z\vert \le 1,\,z\ne 1\). In this paper, we derive new expansions in terms of rational functions of z that converge in different regions, bounded or unbounded, of the complex plane that contain the indented closed unit disk. We give either explicit formulas for the coefficients of the expansions or recurrence relations. The key point of the analysis is the use of multi-point Taylor expansions in appropriate integral representations of \({}_{p+1}F_p(a,b_1,\ldots ,b_p;c_1,\ldots ,c_p;z)\). We show the accuracy of the approximations by means of several numerical experiments.