Computation of Nevanlinna characteristic functions derived from generating functions of some special numbers

Abstract

In the present paper, firstly we find a number of poles of generating functions of Bernoulli numbers and associated Euler numbers, denoted by $n(a,B)$ and $n(a,E)$ , respectively. Secondly, we derive the mean value of a positive logarithm of generating functions of Bernoulli numbers and associated Euler numbers shown as $m(2\pi ,B)$ and $m(\pi ,E)$ , respectively. From these properties, we find Nevanlinna characteristic functions which we stated in the paper. Finally, as an application, we show that the generating function of Bernoulli numbers is a normal function.

Keywords: Associated Euler numbers; Bernoulli numbers; Generating function; Meromorphic function; Nevanlinna characteristic function; Normal function; Poisson–Jensen formula.