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Numerical methods for the computation of the confluent and Gauss hypergeometric functions. (English) Zbl 1360.33009

In this review paper, the authors summarize and overview several methods to compute the confluent hypergeometric function \(\mathbf{M}(a;b,c)\) and Gauß’s hypergeometric function \(\mathbf{F}(a;b,c)\). To this end, they discuss the choice of reliable methods for different parameter and variable regimes. These methods include Taylor series, asymptotic expansions, continued fractions, recurrence relationships, and hyperasymptotic expansions. A publicly available MATLAB code for computing the functions \(\mathbf{M}(a;b,c)\) and \(\mathbf{F}(a;b,c)\) was also developed.

MSC:

33C05 Classical hypergeometric functions, \({}_2F_1\)
33C15 Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\)
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
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