Article ID: MTJPAM-D-25-00241

Title: On a family of probabilistic Frobenius-Euler type Simsek numbers and polynomials with an analysis of their generating functions

Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-25-00241; Volume 7 / Issue 3 / Year 2025, Pages 88-101

Document Type: Research Paper

Author(s): ‪Neslihan Kilar ^a

^aDepartment of Computer Technologies, Bor Vocational School, Niğde Ömer Halisdemir University, TR-51700 Niğde, Turkey

Received: 1 October 2025, Accepted: 8 December 2025, Published: 22 March 2026

Corresponding Author: Neslihan Kilar (Email address: neslihankilar@ohu.edu.tr; neslihankilar@gmail.com)

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Abstract

The main goal of this paper is to introduce the generating functions of probabilistic Frobenius-Euler type Simsek numbers and polynomials associated with Y. By using the generating functions of these numbers and polynomials, many identities and relations are derived. These results are related to Apostol type numbers and polynomials, the Bernoulli numbers and polynomials of the second kind, the probabilistic Stirling numbers of the second kind, the probabilistic Fubini polynomials, the probabilistic Bernoulli polynomials and numbers. Moreover, connections with certain discrete probability distributions are examined through the use of specially defined random variables. These probabilistic interpretations provide a deeper insight into the structure and properties of the defined numbers and polynomials. Finally, several special cases and their applications are presented.

Keywords: Apostol type numbers and polynomials, Bernoulli numbers and polynomials of the second kind, Stirling numbers, Frobenius-Euler type Simsek numbers and polynomials, probabilistic Stirling numbers of the second kind, probabilistic Bernoulli polynomials and numbers

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