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

Abstract:

We consider the confluent hypergeometric function $M(a,b;z)$ for $z\in \mathbb {C}$ and $\Re b>\Re a>0$, and the confluent hypergeometric function $U(a,b;z)$ for $b\in \mathbb {C}$, $\Re a>0$, and $\Re z>0$. We derive two convergent expansions of $M(a,b;z)$; one of them in terms of incomplete gamma functions $\gamma (a,z)$ and another one in terms of rational functions of $e^z$ and $z$. We also derive a convergent expansion of $U(a,b;z)$ in terms of incomplete gamma functions $\gamma (a,z)$ and $\Gamma (a,z)$. The expansions of $M(a,b;z)$ hold uniformly in either $\Re z\ge 0$ or $\Re z\le 0$; the expansion of $U(a,b;z)$ holds uniformly in $\Re z>0$. The accuracy of the approximations is illustrated by means of some numerical experiments.