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New moment formulas for moments and characteristic function of the geometric distribution in terms of Apostol-Bernoulli polynomials and numbers
  • Buket Simsek
Buket Simsek
Akdeniz Universitesi

Corresponding Author:[email protected]

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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 theis characteristic function, the Apostol-Bernoulli polynomials and numbers, and also trigonometric functions including sin w and cos w . 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.
29 Jan 2024Submitted to Mathematical Methods in the Applied Sciences
29 Jan 2024Assigned to Editor
29 Jan 2024Submission Checks Completed
21 Feb 2024Reviewer(s) Assigned
02 Mar 2024Review(s) Completed, Editorial Evaluation Pending
05 Mar 2024Editorial Decision: Accept