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.

Kummer function, numerical computation
Computation of special functions, construction of tables (msc 65D20), Confluent hypergeometric functions, Whittaker functions, ${}_1F_1$ (msc 33C15), Gamma, beta and polygamma functions (msc 33B15), Computation of special functions, construction of tables (msc 65D20)
Null option (theme 11)
Elsevier
Applied Mathematics and Computation

Gil, A, Segura, J, & Temme, N.M. (2015). Computing the Kummer function $U(a,b,z)$ for small values of the arguments. Applied Mathematics and Computation, (271), 532–539.