Probabilistic degenerate Bernoulli and degenerate Euler polynomials
L. Luo, Taekyun Kim, Dae San Kim, Yuankui Ma
Abstract
Recently, many authors have studied degenerate Bernoulli and degenerate Euler polynomials. Let $Y$Y be a random variable whose moment generating function exists in a neighbourhood of the origin. The aim of this paper is to introduce and study the probabilistic extension of degenerate Bernoulli and degenerate Euler polynomials, namely the probabilistic degenerate Bernoulli polynomials associated with $Y$Y and the probabilistic degenerate Euler polynomials associated with $Y$Y. Also, we intoduce the probabilistic degenerate $r$r-Stirling numbers of the second associated with $Y$Y and the probabilistic degenerate two variable Fubini polynomials associated with $Y$Y. We obtain some properties, explicit expressions, recurrence relations and certain identities for those polynomials and numbers. As special cases of $Y$Y, we treat the gamma random variable with parameters $\alpha , \beta \gt 0$α, β>0, the Poisson random variable with parameter $\alpha \gt 0$α>0, and the Bernoulli random variable with probability of success $p$p.