## Asymptotic expansions for the incomplete gamma function in the transition regions

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Gergő Nemes and Adri B. Olde Daalhuis
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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 $\gamma ^*(a,x)$.## References

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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
- MR Author ID: 293428
- Email: a.oldedaalhuis@ed.ac.uk
- 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.
- © Copyright 2018 American Mathematical Society
- Journal: Math. Comp.
**88**(2019), 1805-1827 - MSC (2010): Primary 33B20, 41A60
- DOI: https://doi.org/10.1090/mcom/3391
- MathSciNet review: 3925486