A new addition theorem for cylinder functions

Abstract

We prove a new addition theorem for cylinder functions. The value of an arbitrary zero order cylinder function for an argument x−y with complex-valued x and y is expressed by an infinite series whose summands involve the product of cylinder functions, one of these functions taken at x and the other at y. Other as in the hitherto known addition theorems for cylinder functions, the summands of this series also contain factors that are decreasing powers of x and increasing powers of y, respectively. As a consequence, our addition theorem directly leads to asymptotic expansions at infinity for convolutions of compactly supported functions with zero order cylinder functions.