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Numerically satisfactory solutions of hypergeometric recursions

Authors: Amparo Gil, Javier Segura and Nico M. Temme
Journal: Math. Comp. 76 (2007), 1449-1468
MSC (2000): Primary 33C05, 39A11, 41A60, 65D20
Published electronically: January 31, 2007
MathSciNet review: 2299782
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Abstract: Each family of Gauss hypergeometric functions \[ f_n={}_2F_1(a+\varepsilon _1n, b+\varepsilon _2n ;c+\varepsilon _3n; z), n\in {\mathbb Z} , \] for fixed $\varepsilon _j=0,\pm 1$ (not all $\varepsilon _j$ equal to zero) satisfies a second order linear difference equation of the form \[ A_nf_{n-1}+B_nf_n+C_nf_{n+1}=0. \] Because of symmetry relations and functional relations for the Gauss functions, many of the 26 cases (for different $\varepsilon _j$ values) can be transformed into each other. In this way, only with four basic difference equations can all other cases be obtained. For each of these recurrences, we give pairs of numerically satisfactory solutions in the regions in the complex plane where $|t_1|\neq |t_2|$, $t_1$ and $t_2$ being the roots of the characteristic equation.

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Additional Information

Amparo Gil
Affiliation: Departamento de Matemáticas, Estadística y Computación, Univ. Cantabria, 39005-Santander, Spain

Javier Segura
Affiliation: Departamento de Matemáticas, Estadística y Computación, Univ. Cantabria, 39005-Santander, Spain
MR Author ID: 627158

Nico M. Temme
Affiliation: CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands

Keywords: Gauss hypergeometric functions, recursion relations, difference equations, stability of recursion relations, numerical evaluation of special functions, asymptotic analysis.
Received by editor(s): October 18, 2005
Received by editor(s) in revised form: February 2, 2006
Published electronically: January 31, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.