Probabilistic degenerate Bernoulli and degenerate Euler polynomials

ABSTRACTRecently, many authors have studied degenerate Bernoulli and degenerate Euler polynomials. Let [Formula: see text] 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 [Formula: see text] and the probabilistic degenerate Euler polynomials associated with [Formula: see text]. Also, we intoduce the probabilistic degenerate [Formula: see text]-Stirling numbers of the second associated with [Formula: see text] and the probabilistic degenerate two variable Fubini polynomials associated with [Formula: see text]. We obtain some properties, explicit expressions, recurrence relations and certain identities for those polynomials and numbers. As special cases of [Formula: see text], we treat the gamma random variable with parameters [Formula: see text], the Poisson random variable with parameter [Formula: see text], and the Bernoulli random variable with probability of success [Formula: see text].