Probabilistic poly-Bernoulli numbers

Assume that is Y a random variable whose moment generating function exists in a neighbourhood of the origin. The aim of this paper is to study probabilistic poly-Bernoulli numbers associated with Y, as probabilistic extensions of poly-Bernoulli numbers. We derive explicit expressions, some related identities and a symmetric relation for those numbers. We also investigate explicit expressions for the modified probabilisitc Bernoulli numbers associated with Y, which are slightly different from probabilisitic Bernoulli numbers associated with Y. As special cases of Y, we treat the Poisson, gamma and Bernoulli random variables.