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

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.Comment: To appear in Advances in Dynamical Systems and Applications. Proceedings of the International Conference on Differential Equations, Difference Equations and Special Functions, Patras, Greece, September 3 - 9, 2012, dedicated to the memory of Panayiotis D. Siafarikas. 13 page