Some Identities of the Probabilistic Changhee Polynomials and Their Applications

Special numbers and polynomials are very important tools in diverse fields such as mathematics, physics, engineering, science, and related disciplines, addressing problems in areas like mathematical physics, numerical analysis, differential equations, fluid dynamics, and quantum mechanics. Let Y be a random variable with the moment-generating function of Y. The purpose of this paper is to study the probabilistic Changhee polynomials associated with Y, which generalize the classical Changhee polynomials using probabilistic methods. We investigate the relationships between these polynomials and the probabilistic Stirling numbers of the first and second kinds, the Changhee polynomials, the Stirling numbers of the second kind, the falling factorial sequence, probabilistic Euler polynomials, and Fubini polynomials. Additionally, several interesting identities are derived based on these relationships, and we present the graphs and root distributions of the probabilistic Changhee polynomials associated with Y using Mathematica.