On the Nevanlinna characteristic of confluent hypergeometric functions

ABSTRACT

A confluent hypergeometric function (Kummer's function) is a generalized hypergeometric series introduced by Kummer in 1837 [De integralibus quibusdam definitis et seriebus infinitis. J Reine Angew Math (in Latin). 1837;17:228–242], given by $M(\alpha ;\gamma ;z)={}_1{F}_1(\alpha ;\gamma ;z):=\sum _{n=0}^{\mathrm{\infty }}\left((\alpha {)}_n/n!(\gamma {)}_n\right)z^n(\gamma \ne 0,-1,-2,\dots )$, which are of great applications in statistics, mathematical physics, engineering and so on. In this paper, we investigate some properties of Kummer's function from viewpoint 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 degenerated and non-degenerated cases respectively. Moreover, we obtain an asymptotic estimate of characteristic function $T(r,M(\alpha ;\gamma ;z\left)\right)$ and calculate the logarithmic derivative $m(r,M^{\prime }(\alpha ;\gamma ;z)/M(\alpha ;\gamma ;z))$, the distribution of zeros of Kummer's function is also discussed. Finally, we show Kummer's function and an entire function are uniquely determined by their c-values.

KEYWORDS:

• Kummer's function
• Nevanlinna theory
• Wiman-Valiron theory

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• Primary: 33C15
• 30D30
• 30D35

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Xu-Dan Luo http://orcid.org/0000-0002-3833-5530

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (11371225) and the Natural Science Foundation of Fujian Province in China (2011J01006).