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

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 1F1-functions needs a limiting procedure. We use the power series of the 1F1-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′(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.