Novel formulas of moments of Negative Binomial distribution connected with Apostol-Bernoulli numbers of higher order and Stirling numbers

The main subject of this article is to present and reveal some new relationships between the moment generating functions of the Negative Binomial distribution and the generating functions for the Apostol-Bernoulli numbers and polynomials. By the help of these relations and Binomial series, we derive many computation formulas. These formulas give relations among moments, factorial moments, and the Apostol-Bernoulli numbers and polynomials, the Stirling numbers, and also other special functions related to zeta functions. By using these formulas, we give some numerical values of moments, expected value, and variance. Finally, we give some observations on formulas for the moments involivin binomial series and zeta functions.