Some Identities of the probabilistic higher order frobenius-euler polynomials and their applications

Recently, there has been significant research on the generalization of various numbers using probabilistic methods. In this paper, we introduce the probabilistic higher order Frobenius-Euler polynomials which are a generalization of Frobenius-Euler polynomials using probabilistic methods and demonstrate that these polynomials can be expressed as a linear combination of probabilistic Stirling numbers of the second kind and falling factorial sequences which provide both practical applications and profound insights into these polynomials. Furthermore, we show that when [Formula: see text] is a Bernoulli, gamma, or Poisson random variable, it can be expressed as a linear combination of the Stirling numbers of the second kind, Bell polynomials, or rising factorial sequences, respectively, and derive several new and interesting identities of those polynomials. We also investigate properties of the polynomials [Formula: see text] using graphs for different values of the random variable [Formula: see text], two different integers [Formula: see text], and three different integers [Formula: see text].