New moment formulas for moments and characteristic function of the geometric distribution in terms of Apostol-Bernoulli polynomials and numbers

Although it is very easy to calculate the 1st moment and 2nd moment values of the geometric distribution with the methods available in existing books and other articles, it is quite difficult to calculate moment values larger than the 3rd order. Because in order to find these moment values, many higher order derivatives of the geometric series and convergence properties of the series are needed. The aim of this article is to find new formulas for characteristic function of the geometric random variable (with parameter p) in terms of the Apostol-Bernoulli polynomials and numbers, and the Stirling numbers. This characteristic function characterizes the geometric distribution. Using the Euler's identity, we give relations among this characteristic function, the Apostol-Bernoulli polynomials and numbers, and also trigonometric functions including cosine$$ cosine $$ and sine$$ sine $$. A relations between the characteristic function and the moment generating function is also given. By using these relations, we derive new moments formulas in terms of the Apostol-Bernoulli polynomials and numbers. Moreover, we give some applications of our new formulas.