Explicit Formulas for Probabilistic Multi-Poly-Bernoulli Polynomials and Numbers

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 study probabilistic Bernoulli polynomials of order $$r$$ associated with $$Y$$ and probabilistic multi-poly-Bernoulli polynomials associated with $$Y$$ . They are respectively probabilistic extensions of Bernoulli polynomials of order $$r$$ and multi-poly-Bernoulli polynomials. We find explicit expressions, certain related identities and some properties for them. In addition, we treat the special cases of Poisson, gamma and Bernoulli random variables. DOI 10.1134/S1061920824030087

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