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Numerical evaluation of the Gauss hypergeometric function by power summations


Author: Jurgen A. Doornik
Journal: Math. Comp. 84 (2015), 1813-1833
MSC (2010): Primary 33C05, 65D20
Published electronically: December 3, 2014
MathSciNet review: 3335893
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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 $ \vert z-0.32\vert > 0.32$ for $ {\mathcal {R}}z>0$, which is tighter than the $ \vert z-0.5\vert > 0.5$ which is normally considered. Finally, we derive an improved algorithm for the numerical evaluation of $ {}_2F_1$.


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Additional Information

Jurgen A. Doornik
Affiliation: Institute for New Economic Thinking at the Oxford Martin School, University of Oxford
Email: jurgen.doornik@nuffield.ox.ac.uk

DOI: https://doi.org/10.1090/S0025-5718-2014-02905-0
Keywords: Gauss hypergeometric function
Received by editor(s): May 1, 2013
Received by editor(s) in revised form: September 25, 2013, and October 25, 2013
Published electronically: December 3, 2014
Additional Notes: This research was supported in part by grants from the Open Society Foundations and the Oxford Martin School.
Article copyright: © Copyright 2014 American Mathematical Society