Some new properties of Confluent Hypergeometric Functions

Abstract

The confluent hypergeometric functions (the Kummer functions) defined by ${}_1{F}_1(\alpha ;\gamma ;z):={\sum }_{n=0}^{\infty }\frac{(\alpha {)}_n}{n!(\gamma {)}_n}{z}^n(\gamma \ne 0,-1,-2,\cdots )$, 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 $\alpha \in {\mathbb{Z}}_{\le 0}$ and $\alpha \notin {\mathbb{Z}}_{\le 0}$, 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 $m\left(r,\frac{{}_1{F}_1^{\prime }(\alpha ;\gamma ;z)}{{}_1{F}_1(\alpha ;\gamma ;z)}\right)$, 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.

arXiv comment20 pages. Submitted to Journal of Mathematical Analysis and Applications