Numerical Algorithms

, Volume 74, Issue 3, pp 821–866 | Cite as

Numerical methods for the computation of the confluent and Gauss hypergeometric functions

  • John W. PearsonEmail author
  • Sheehan Olver
  • Mason A. Porter
Original Paper


The two most commonly used hypergeometric functions are the confluent hypergeometric function and the Gauss hypergeometric function. We review the available techniques for accurate, fast, and reliable computation of these two hypergeometric functions in different parameter and variable regimes. The methods that we investigate include Taylor and asymptotic series computations, Gauss–Jacobi quadrature, numerical solution of differential equations, recurrence relations, and others. We discuss the results of numerical experiments used to determine the best methods, in practice, for each parameter and variable regime considered. We provide “roadmaps” with our recommendation for which methods should be used in each situation.


Computation of special functions Confluent hypergeometric function Gauss hypergeometric function 

Mathematics Subject Classification (2010)

Primary: 33C05 33C15 Secondary: 41A58 41A60 


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • John W. Pearson
    • 1
    Email author
  • Sheehan Olver
    • 2
  • Mason A. Porter
    • 3
  1. 1.School of Mathematics, Statistics and Actuarial ScienceUniversity of KentCanterburyUK
  2. 2.School of Mathematics and StatisticsThe University of SydneySydneyAustralia
  3. 3.Oxford Centre for Industrial and Applied Mathematics, Mathematical InstituteUniversity of OxfordWoodstock RoadUK

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