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 α,β > 0 , the Poisson random variable with parameter α >0 , and the Bernoulli random variable with probability of success p . DOI 10.1134/S106192084010072