Convergence of Magnus integral addition theorems for confluent hypergeometric 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.

KEYWORDS:

• Confluent hypergeometric functions
• Bessel functions
• modified parabolic cylinderfunctions
• integral addition theorems

AMS SUBJECT CLASSIFICATION:

• 33C10
• 33C15

Acknowledgments

We thank Adri Olde Daalhuis and Nico Temme for valuable discussions.