Numerical evaluation of the Gauss hypergeometric function by power summations
Jurgen A. Doornik · Mathematics of Computation · 2014
Numerical evaluation of the Gauss hypergeometric function 2 F 1 ( a , b ; c ; z ) {}_2F_1(a,b;c;z) , with complex parameters a , b , c a,b,c and complex argument z z is notoriously difficult. Carrying out the summation that defines the function may fail, even for moderate values of z 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 b-a or c − a − b c-a-b is an integer, and suffer numerical instability near that. This singularity has to be removed analytically after collecting powers in z z . The contributions in this paper are fourfold. First, analytical expressions are provided that remove the singularity from Bühring’s 1 / ( z − z 0 ) 1/(z-z_0) transformation. This is more difficult, because the singularity occurs twice, and it is necessary to collect powers of z 0 z_0 , as well as z 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 |z-0.32| > 0.32 for