Convergence of Magnus integral addition theorems for confluent hypergeometric functions

TL;DR

This paper analyzes the convergence domains of Magnus' integral addition theorems for confluent hypergeometric functions and derives new addition formulas for related special functions using asymptotic analysis.

Contribution

It determines convergence domains for Magnus' addition theorem and extends results to various special functions through asymptotic methods.

Findings

Established precise convergence domains for Magnus' addition theorem.

Derived new integral addition theorems for related special functions.

Connected asymptotic analysis with integral representations of special functions.

Abstract

In 1946, Magnus presented an addition theorem for the confluent hypergeometric function of the second kind $U$ with argument $x+y$ expressed as an integral of a product of two $U$'s, one with argument $x$ and another with argument $y$. We take advantage of recently obtained asymptotics for $U$ with large complex first parameter to determine a domain of convergence for Magnus' result. Using well-known specializations of $U$, we obtain corresponding integral addition theorems with precise domains of convergence for modified parabolic cylinder functions, and Hankel, Macdonald, and Bessel functions of the first and second kind with order zero and one.