On the zeros of Confluent Hypergeometric Functions

Abstract

In this paper, we study the zero sets of the confluent hypergeometric function ${}_1{F}_1(\alpha ;\gamma ;z):={\sum }_{n=0}^{\infty }\frac{(\alpha {)}_n}{n!(\gamma {)}_n}{z}^n$, where $\alpha ,\gamma ,\gamma -\alpha \notin {\mathbb{Z}}_{\le 0}$, and show that if $\{z_n{\}}_{n=1}^{\infty }$ is the zero set of ${}_1{F}_1(\alpha ;\gamma ;z)$ with multiple zeros repeated and modulus in increasing order, then there exists a constant $M>0$ such that $|z_n|\ge Mn$ for all $n\ge 1$.