Complete systems of Tricomi functions in spaces of holomorphic functions

Abstract

Let $\Psi(a,c;z)$ be the main branch of Tricomi confluent hypergeometric function with parameters $a,c$ and $G$ be an arbitrary simply connected subregion of the complex plane cut along the real non-positive semiaxis. It is proved that a system of the kind

\[\big\{\Psi(n + \lambda + \alpha + 1, \alpha + 1; z)\big\}_{n=0}^{\infty}\]

is complete in the space of the complex functions holomorphic in $G$ provided that $\lambda$ and $\alpha$ are real and $\lambda + \alpha > -1$.