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arXiv · 1510.01285

On the zeros of Confluent Hypergeometric Functions

Wei-Chuan Lin·Xu-Dan Luo
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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 $n\geq 1$.

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Wei-Chuan Lin, Xu-Dan Luo. 2015-10-05. On the zeros of Confluent Hypergeometric Functions. https://arxiv.org/abs/1510.01285

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