Probabilistic Type 2 poly-Bernoulli Polynomials

The main purpose of this article is to introduce the probabilistic type 2 poly-Bernoulli polynomials under the condition that Y is a random variable. This means that we will consider the probabilistic extension of the type 2 poly-Bernoulli polynomials and study to obtain some new results. Furthermore, we also define the probabilistic unipoly-Bernoulli polynomials and numbers attached to p, and investigate their interesting basic properties. Based on these new definition, we derive some meaningful formulae of probabilistic type 2 poly-Bernoulli polynomials and probabilistic unipoly-Bernoulli polynomials and numbers attached to p.