Asymptotic expansions of Kummer hypergeometric functions for large values of the parameters

TL;DR

This paper develops uniform asymptotic expansions for Kummer hypergeometric functions when parameters are large, covering various ratios of the parameters, which enhances understanding of their behavior in asymptotic regimes.

Contribution

The paper introduces a unified method to derive asymptotic expansions of Kummer functions for large parameters, including the case where parameters are comparable.

Findings

Derived asymptotic expansions for large parameters a and b

Handled cases with different ratios b/a, including a~b

Provided uniform approximations valid across parameter regimes

Abstract

We derive asymptotic expansions of the Kummer functions $M(a,b,z)$ and $U(a,b+1,z)$ for large positive values of $a$ and $b$, with $z$ fixed. For both functions we consider $b/a\le 1$ and $b/a\ge 1$, with special attention for the case $a\sim b$. We use a uniform method to handle all cases of these parameters.