Probabilistic Bernoulli and Euler Polynomials
T. Kim, Dae San Kim
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 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 $$p$$ . DOI 10.1134/S106192084010072