Asymptotic expansions of Kummer hypergeometric functions with three asymptotic parameters $a$, $b$ and $z$

TL;DR

This paper extends asymptotic expansions of Kummer hypergeometric functions to include large values of parameters a, b, and z, providing more comprehensive approximations with numerical validation.

Contribution

It introduces new asymptotic expansions for Kummer functions accommodating large a, b, and z, expanding previous work which focused on large a and b with fixed z.

Findings

New asymptotic expansions valid for large a, b, or z

Numerical tables demonstrating the accuracy of the expansions

Extension of previous methods to include large z values

Abstract

In a recent paper \cite{Temme:2021:AKH} new asymptotic expansions are given for the Kummer functions $M(a,b,z)$ and $U(a,b+1,z)$ for large positive values of $a$ and $b$, with $z$ fixed and special attention for the case $a\sim b$. In this paper we extend the approach and also accept large values of $z$. The new expansions are valid when at least one of the parameters $a$, $b$, or $z$ is large. We provide numerical tables to show the performance of the expansions.