Some new properties of Confluent Hypergeometric Functions

Abstract: The confluent hypergeometric functions (the Kummer functions) defined by ${}{1}F{1}(\alpha;\gamma;z):=\sum_{n=0}^{\infty}\frac{(\alpha){n}}{n!(\gamma){n}}z^{n}\ (\gamma\neq 0,-1,-2,\cdots)$, 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.