Probabilistic degenerate Dowling polynomials associated with random variables

Abstract

The aim of this paper is to study probabilistic versions of the degenerate Whitney numbers of the second kind and those of the degenerate Dowling polynomials, namely the probabilistic degenerate Whitney numbers of the second kind associated with Y and the probabilistic degenerate Dowling polynomials associated with Y. Here Y is a random variable whose moment generating function exists in some neighborhood of the origin. We derive some properties, explicit expressions, certain identities, recurrence relations and generating functions for those numbers and polynomials. In addition, we investigate their generalizations, namely the probabilistic degenerate r-Whitney numbers of the second kind associated with Y and the probabilistic degenerate r-Dowling polynomials associated with Y, and get similar results to the aforementioned numbers and polynomials.