Computing the Kummer function U(a,b,z) for small values of the arguments

Abstract

We describe methods for computing the Kummer function $U(a,b,z)$ for small values of $z$, with special attention to small values of $b$. For these values of $b$ the connection formula that represents $U(a,b,z)$ as a linear combination of two ${}_1{F}_1$-functions needs a limiting procedure. We use the power series of the ${}_1{F}_1$-functions and consider the terms for which this limiting procedure is needed. We give recursion relations for higher terms in the expansion, and we consider the derivative $U^{\prime }(a,b,z)$ as well. We also discuss the performance for small $|z|$ of an asymptotic approximation of the Kummer function in terms of modified Bessel functions.

Accepted for publication in Appl. Math. Comput