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Fast and stable rational approximation of generalized hypergeometric functions

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Abstract

Rational approximations of generalized hypergeometric functions \({}_pF_q\) of type \((n+k,k)\) are constructed by the Drummond and factorial Levin-type sequence transformations. We derive recurrence relations for these rational approximations that require \(\mathcal {O}[\max \{p,q\}(n+k)]\) flops. These recurrence relations come in two forms: for the successive numerators and denominators; and, for an auxiliary rational sequence and the rational approximations themselves. Numerical evidence suggests that these recurrence relations are much more stable than the original formulæ for the Drummond and factorial Levin-type sequence transformations. Theoretical results on the placement of the poles of both transformations confirm the superiority of factorial Levin-type transformation over the Drummond transformation.

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Availability of supporting data

All data generated or analyzed through the Hypergeometric Func-tions.jl package during this study are included in this published article. Source code for the figures are available on reasonable request.

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Acknowledgements

We thank Nick Trefethen for a discussion on Padé approximation to \(e^z\). RMS is supported by the Natural Sciences and Engineering Research Council of Canada, through a Discovery Grant (RGPIN-2017-05514).

Funding

RMS is supported by the Natural Sciences and Engineering Research Council of Canada, through a Discovery Grant (RGPIN-2017-05514).

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  1. Department of Mathematics, University of Manitoba, Winnipeg, Canada

    Richard Mikaël Slevinsky

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R.M.S. wrote the manuscript and the code.

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Correspondence to Richard Mikaël Slevinsky.

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Slevinsky, R.M. Fast and stable rational approximation of generalized hypergeometric functions. Numer Algor (2024). https://doi.org/10.1007/s11075-024-01808-w

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