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

TL;DR

This paper develops methods for accurately computing the Kummer function U(a,b,z) when the arguments are small, especially for small b, using power series, recursion, and asymptotic approximations.

Contribution

It introduces new techniques for computing U(a,b,z) for small arguments, including handling limiting procedures and recursion relations, improving numerical stability.

Findings

Derived recursion relations for higher-order terms.

Analyzed the performance of asymptotic approximations.

Provided methods for computing derivatives of U(a,b,z).

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 ${}_1F_1$-functions needs a limiting procedure. We use the power series of the ${}_1F_1$-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 $\vert z\vert$ of an asymptotic approximation of the Kummer function in terms of modified Bessel functions.