## Numerical evaluation of the Gauss hypergeometric function by power summations

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- by Jurgen A. Doornik PDF
- Math. Comp.
**84**(2015), 1813-1833 Request permission

## 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 ${}_2F_1$.

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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
- 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.
- © Copyright 2014 American Mathematical Society
- Journal: Math. Comp.
**84**(2015), 1813-1833 - MSC (2010): Primary 33C05, 65D20
- DOI: https://doi.org/10.1090/S0025-5718-2014-02905-0
- MathSciNet review: 3335893