Representations by probabilistic Bernoulli and degenerate Bernoulli polynomials

TL;DR

The paper addresses representing an arbitrary polynomial $p(x)$ as a linear combination of probabilistic Bernoulli polynomials $B_n^{Y}(x)$ and probabilistic degenerate Bernoulli polynomials $\beta_{n,\lambda}^{Y}(x)$ tied to a random variable $Y$ with an analytic moment generating function. It develops an umbral calculus framework to derive explicit coefficient formulas, expressed via forward differences and probabilistic Stirling numbers $S_1^{Y}, S_2^{Y}$ (and their degenerate variants), and extends these results to higher-order polynomials $B_n^{Y,(r)}(x)$ and $\beta_{n,\lambda}^{Y,(r)}(x)$. The authors provide detailed representations for $p(x)$ in terms of both first- and higher-order probabilistic Bernoulli families, and illustrate the approach with numerous distributional examples, including the classical case $Y=1$ that recovers known identities. This framework unifies stochastic and umbral methods to enable systematic polynomial decompositions across a wide range of discrete and continuous distributions.

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.

Theorems & Definitions (12)

• Proposition 1.1
• Proposition 1.2
• Theorem 3.1
• Remark 3.2
• Theorem 3.3
• Remark 3.4
• Theorem 4.1
• Theorem 4.2
• Remark 4.3
• Lemma 5.1