Representations by probabilistic Bernoulli and degenerate Bernoulli polynomials

TL;DR

This paper explores how to represent arbitrary polynomials using probabilistic Bernoulli and degenerate Bernoulli polynomials linked to a random variable, employing umbral calculus for explicit formulas.

Contribution

It introduces explicit formulas for polynomial representations via probabilistic Bernoulli polynomials using umbral calculus, extending to higher-order cases and various random variables.

Findings

Explicit formulas for polynomial representations derived

Representation methods applicable to discrete and continuous variables

Extension to higher-order polynomial representations

Abstract

We investigate the representation of arbitrary polynomials using probabilistic Bernoulli and degenerate Bernoulli polynomials associated with a random variable $Y$, whose moment generating function exists in a neighborhood of the origin. In addition, this paper explores the problem of representing arbitrary polynomials in terms of their higher-order counterparts. We develop explicit formulas for those representations with the help of umbral calculus and illustrate our results for several discrete and continuous random variables Y.