Numerical evaluation of the Gauss hypergeometric function by power summations

Abstract:

Numerical evaluation of the Gauss hypergeometric function ${}_2F_1(a,b;c;z)$, with complex parameters $a,b,c$ and complex argument $z$ is notoriously difficult. Carrying out the summation that defines the function may fail, even for moderate values of $z$. Formulae are available to transform the effective argument in the series, potentially leading to a numerically successful summation. Unfortunately, these transformations have a singularity when $b-a$ or $c-a-b$ is an integer, and suffer numerical instability near that. This singularity has to be removed analytically after collecting powers in $z$.

The contributions in this paper are fourfold. First, analytical expressions are provided that remove the singularity from Bühring’s $1/(z-z_0)$ transformation. This is more difficult, because the singularity occurs twice, and it is necessary to collect powers of $z_0$, as well as $z$. The resulting expression has a three-term recursion, like the original. Next, improved expressions are derived for the cases that have been addressed before. We study a transformation that converges outside $|z-0.32| > 0.32$ for ${\mathcal {R}}z>0$, which is tighter than the $|z-0.5| > 0.5$ which is normally considered. Finally, we derive an improved algorithm for the numerical evaluation of ${}_2F_1$.