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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Numerically satisfactory solutions of hypergeometric recursions
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by Amparo Gil, Javier Segura and Nico M. Temme PDF
Math. Comp. 76 (2007), 1449-1468 Request permission


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
  • Email:
  • Javier Segura
  • Affiliation: Departamento de Matemáticas, Estadística y Computación, Univ. Cantabria, 39005-Santander, Spain
  • MR Author ID: 627158
  • Email:
  • Nico M. Temme
  • Affiliation: CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands
  • Email:
  • Received by editor(s): October 18, 2005
  • Received by editor(s) in revised form: February 2, 2006
  • Published electronically: January 31, 2007
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 76 (2007), 1449-1468
  • MSC (2000): Primary 33C05, 39A11, 41A60, 65D20
  • DOI:
  • MathSciNet review: 2299782