Complete systems of Tricomi functions in spaces of holomorphic functions

Abstract

Let $\mathrm{\Psi }(a,c;z)$ 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

$$\{\mathrm{\Psi }(n+\lambda +\alpha +1,\alpha +1;z){\}}_{n=0}^{\mathrm{\infty }}$$

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