Convergent expansions of the confluent hypergeometric functions in terms of elementary functions

Abstract

We consider the confluent hypergeometric function M(a, b; z) for z ∈ C and Rb >Ra > 0, and the confluent hypergeometric function U(a, b; z) for b ∈ C, Ra > 0, and Rz > 0. We derive two convergent expansions of M(a, b; z); one of them in terms of incomplete gamma functions γ(a, z) and another one in terms of rational functions of ez and z. We also derive a convergent expansion of U(a, b; z) in terms of incomplete gamma functions γ(a, z) and Γ(a, z). The expansions of M(a, b; z) hold uniformly in either Rz ≥ 0 or Rz ≤ 0; the expansion of U(a, b; z) holds uniformly in Rz > 0. The accuracy of the approximations is illustrated by means of some numerical experiments.