The Asymptotic Expansion of Kummer Functions for Large Values of the $a$-Parameter, and Remarks on a Paper by Olver

TL;DR

This paper extends the known asymptotic expansion of the Kummer function U(a,b,z) to the entire Riemann surface of the logarithm for large a, and generalizes Olver's earlier results.

Contribution

It proves the validity of the asymptotic expansion of Kummer functions on the full Riemann surface and in a broader setting than previously established.

Findings

Asymptotic expansion valid on the full Riemann surface of the logarithm

Generalization of Olver's results to a wider setting

Enhanced understanding of Kummer functions for large parameters

Abstract

It is shown that a known asymptotic expansion of the Kummer function $U(a,b,z)$ as $a$ tends to infinity is valid for $z$ on the full Riemann surface of the logarithm. A corresponding result is also proved in a more general setting considered by Olver (1956).