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Asymptotic expansions for the incomplete gamma function in the transition regions


Authors: Gergő Nemes and Adri B. Olde Daalhuis
Journal: Math. Comp. 88 (2019), 1805-1827
MSC (2010): Primary 33B20, 41A60
DOI: https://doi.org/10.1090/mcom/3391
Published electronically: November 8, 2018
MathSciNet review: 3925486
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Abstract: We construct asymptotic expansions for the normalised incomplete gamma function $ Q(a,z)=\Gamma (a,z)/\Gamma (a)$ that are valid in the transition regions, including the case $ z\approx a$, and have simple polynomial coefficients. For Bessel functions, these types of expansions are well known, but for the normalised incomplete gamma function they were missing from the literature. A detailed historical overview is included. We also derive an asymptotic expansion for the corresponding inverse problem, which has importance in probability theory and mathematical statistics. The coefficients in this expansion are again simple polynomials, and therefore its implementation is straightforward. As a byproduct, we give the first complete asymptotic expansion as $ a\to -\infty $ of the unique negative zero of the regularised incomplete gamma function $ \gamma ^*(a,x)$.


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Additional Information

Gergő Nemes
Affiliation: School of Mathematics, The University of Edinburgh, James Clerk Maxwell Building, The King’s Buildings, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom
Email: gergo.nemes@ed.ac.uk

Adri B. Olde Daalhuis
Affiliation: School of Mathematics, The University of Edinburgh, James Clerk Maxwell Building, The King’s Buildings, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom
Email: a.oldedaalhuis@ed.ac.uk

DOI: https://doi.org/10.1090/mcom/3391
Keywords: Incomplete gamma functions, asymptotic expansions, transition regions
Received by editor(s): March 21, 2018
Received by editor(s) in revised form: May 11, 2018
Published electronically: November 8, 2018
Additional Notes: The authors’ research was supported by a research grant (GRANT11863412/70NANB15H221) from the National Institute of Standards and Technology.
Article copyright: © Copyright 2018 American Mathematical Society