On the zeros of Confluent Hypergeometric Functions

TL;DR

This paper investigates the distribution of zeros of the confluent hypergeometric function, establishing a linear growth bound on the modulus of its zeros under certain conditions.

Contribution

It provides a new result showing that zeros of the confluent hypergeometric function grow at least linearly in modulus, extending understanding of its zero distribution.

Findings

Zeros grow at least linearly in modulus

Zero set is unbounded and well-structured

Provides bounds for 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 $\alpha, \gamma, \gamma-\alpha\not\in \mathbb{Z}_{\leq 0}$, 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 $n\geq 1$.