Some new properties of Confluent Hypergeometric Functions

TL;DR

This paper explores new properties of confluent hypergeometric functions using value distribution theory, including growth orders, zero distribution, and uniqueness results, enhancing understanding of their complex behavior.

Contribution

It introduces novel growth order classifications, asymptotic estimates, and zero distribution insights for confluent hypergeometric functions, extending existing mathematical theory.

Findings

Different growth orders for special parameter cases

Asymptotic estimation of characteristic functions

Distribution of zeros and uniqueness results

Abstract

The confluent hypergeometric functions (the Kummer functions) defined by ${}_{1}F_{1}(\alpha;\gamma;z):=\sum_{n=0}^{\infty}\frac{(\alpha)_{n}}{n!(\gamma)_{n}}z^{n}\ (\gamma\neq 0,-1,-2,\cdots)$, which are of many properties and great applications in statistics, mathematical physics, engineering and so on, have been given. In this paper, we investigate some new properties of ${}_{1}F_{1}(\alpha;\gamma;z)$ from the perspective of value distribution theory. Specifically, two different growth orders are obtained for $\alpha\in \mathbb{Z}_{\leq 0}$ and $\alpha\not\in \mathbb{Z}_{\leq 0}$, which are corresponding to the reduced case and non-degenerated case of ${}_{1}F_{1}(\alpha;\gamma;z)$. Moreover, we get an asymptotic estimation of characteristic function $T(r,{}_{1}F_{1}(\alpha;\gamma;z))$ and a more precise result of $m\left(r, \frac{{}_{1}F_{1}'(\alpha;\gamma;z)}{{}_{1}F_{1}(\alpha;\gamma;z)}\right)$, compared with the Logarithmic Derivative Lemma. Besides, the distribution of zeros of the confluent hypergeometric functions is discussed. Finally, we show how a confluent hypergeometric function and an entire function are uniquely determined by their $c$-values.