Representations by probabilistic Frobenius-Euler and degenerate Frobenius-Euler polynomials

TL;DR

This paper develops a probabilistic framework to represent arbitrary polynomials using Frobenius-Euler and degenerate Frobenius-Euler polynomials linked to a random variable, employing umbral calculus for explicit formulas.

Contribution

It introduces a novel probabilistic approach to polynomial representation via Frobenius-Euler polynomials associated with random variables, including higher-order cases.

Findings

Explicit formulas for polynomial representations derived

Applications demonstrated for various random variables

Extension to higher-order Frobenius-Euler polynomials

Abstract

Let Y be a random variable whose moment generating function exists in a neighborhood of the origin. The aim of this paper is to represent arbitrary polynomials in terms of probabilistic Frobenius-Euler polynomials associated with Y and probabilistic degenerate Frobenius-Euler polynomials associated with Y , and more generally of their higher-order counterparts. We derive explicit formulas with the help of umbral calculus and illustrate our results in the case of several random variables Y