Asymptotic expansions for the incomplete gamma function in the transition regions

TL;DR

This paper develops simple polynomial asymptotic expansions for the normalized incomplete gamma function in transition regions, including the case where z is approximately equal to a, filling a gap in the literature.

Contribution

It introduces the first comprehensive asymptotic expansions for the incomplete gamma function in transition regions and for its inverse, with straightforward polynomial coefficients.

Findings

Derived asymptotic expansions valid near z≈a

Provided a simple polynomial form for inverse expansions

Presented the first asymptotic expansion for the negative zero of the regularized incomplete gamma function

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 type 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 $\gamma^*(a,x)$.