Remarks on Slater's asymptotic expansions of Kummer functions for large values of the $a-$parameter

TL;DR

This paper reviews and compares asymptotic expansions of Kummer functions for large parameters, highlighting derivation methods and discrepancies with classical results from Slater's work.

Contribution

It provides a summary of asymptotic expansions from differential equations and integral representations, revealing differences with Slater's established expansions.

Findings

Derived new asymptotic expansions using integral representations

Identified discrepancies with Slater's classical expansions

Compared methods for large parameter asymptotics

Abstract

In Slater's 1960 standard work on confluent hypergeometric functions, also called Kummer functions, a number of asymptotic expansions of these functions can be found. We summarize expansions derived from a differential equation for large values of the $a-$parameter. We show how similar expansions can be derived by using integral representations, and we observe discrepancies with Slater's expansions.