Analytical and numerical aspects of a generalization of the complementary error function

In this paper we discuss analytical and numerical properties of the function Vν,μ(α,β,z)=∫0∞e-zt(t+α)ν(t+β)μdt, with α,β,Rz>0α,β,Rz>0, which can be viewed as a generalization of the complementary error function, and in fact also as a generalization of the Kummer U-function. The function Vν,μ(α, β, z) is used for certain values of the parameters as an approximate in a singular perturbation problem. We consider the relation with other special functions and give asymptotic expansions as well as recurrence relations. Several methods for its numerical evaluation and examples are given.