Representations by Probabilistic Bernoulli and Degenerate Bernoulli Polynomials

Recent years have seen a significant resurgence of interest in both degenerate versions and probabilistic extensions of special polynomials and numbers. However, traditional extensions often rely on cumulant generating functions whose associated Stirling numbers lack essential algebraic properties, such as orthogonality and inverse relations, complicating inversion-dependent polynomial expansions. To overcome these limitations, this paper utilizes the framework of umbral calculus to systematically investigate the representations of arbitrary polynomials as linear combinations of probabilistic Bernoulli polynomials and probabilistic degenerate Bernoulli polynomials associated with a random variable Y. We introduce novel probabilistic Stirling numbers of the first and second kinds and , alongside their degenerate counterparts, which explicitly preserve crucial orthogonality and inverse relations. Using these algebraic structures, we derive exact, novel coefficient formulas for expressing arbitrary polynomials in terms of both standard and higher-order probabilistic Bernoulli and degenerate Bernoulli polynomials. To illustrate the utility of these results, we compute explicit, closed-form expansions of the monomial xn for several discrete and continuous random variables, including a detailed evaluation using the exponential distribution. Our framework provides a unified and significantly simplified approach to polynomial expansions compared to existing literature.