New series expansions of the 3F2 function

Abstract

We can use the power series definition of 3F2(a1, a2, a3; b1, b2; z) to compute this function for z in the unit disk only. In this paper we obtain new expansions of this function that are convergent in larger domains. Some of these expansions involve the polynomial 3F2(a1,−n, a3; b1, b2; z) evaluated at certain points z. Other expansions involve the Gauss hypergeometric function 2F1. The domain of convergence is sometimes a disk, other times a half-plane, other times the region |z|2 < 4|1 − z|. The accuracy of the approximation given by these expansions is illustrated with numerical experiments.