Probabilistic Bernoulli and Euler Polynomials

Let Y be a random variable whose moment generating function exists in a neighborhood of the origin. The aim of this paper is to introduce and study the probabilistic extension of Bernoulli polynomials and Euler polynomials, namely the probabilistic Bernoulli polynomials associated Y and the probabilistic Euler polynomials associated with Y. Also, we introduce the probabilistic r -Stirling numbers of the second associated Y, the probabilistic two variable Fubini polynomials associated Y, and the probabilistic poly -Bernoulli polynomials associated with Y. We obtain some properties, explicit expressions, certain identities and recurrence relations for those polynomials. As special cases of Y, we treat the gamma random variable with parameters alpha, beta > 0, the Poisson random variable with parameter alpha > 0, and the Bernoulli random variable with probability of success p.