New series expansions of the Gauss hypergeometric function

Abstract

The Gauss hypergeometric function ₂ F ₁(a,b,c;z) can be computed by using the power series in powers of $z, z/(z-1), 1-z, 1/z, 1/(1-z),~\textrm{and}~(z-1)/z$. With these expansions, ₂ F ₁(a,b,c;z) is not completely computable for all complex values of z. As pointed out in Gil et al. (2007, §2.3), the points z = e ^±iπ/3 are always excluded from the domains of convergence of these expansions. Bühring (SIAM J Math Anal 18:884–889, 1987) has given a power series expansion that allows computation at and near these points. But, when b − a is an integer, the coefficients of that expansion become indeterminate and its computation requires a nontrivial limiting process. Moreover, the convergence becomes slower and slower in that case. In this paper, we obtain new expansions of the Gauss hypergeometric function in terms of rational functions of z for which the points z = e ^±iπ/3 are well inside their domains of convergence. In addition, these expansions are well defined when b − a is an integer and no limits are needed in that case. Numerical computations show that these expansions converge faster than Bühring’s expansion for z in the neighborhood of the points e ^±iπ/3, especially when b − a is close to an integer number.