On the zeros of Confluent Hypergeometric Functions (1510.01285v1)

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 $\alpha, \gamma, \gamma-\alpha\not\in \mathbb{Z}{\leq 0}$, 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 $|z_n|\geq M n$ for all $n\geq 1$.