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.

MSC 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)
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.