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arXiv · 1509.06465

Some new properties of Confluent Hypergeometric Functions

Xu-Dan Luo·Wei-Chuan Lin
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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.

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Xu-Dan Luo, Wei-Chuan Lin. 2015-09-22. Some new properties of Confluent Hypergeometric Functions. https://arxiv.org/abs/1509.06465

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