Analytical and Numerical Aspects of a Generalization of the Complementary Error Function

In this paper we discuss analytical and numerical properties of the function $V_{\nu,\mu}(\alpha,\beta,z)=\int_0^\infty e^{-zt}(t+\alpha)^\nu(t+\beta)^\mu\,dt$, with $\alpha,\beta,\Re z>0$, 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 $V_{\nu,\mu}(\alpha,\beta,z)$ is used for certain values of the parameters as an approximant 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.