Probabilistic Degenerate Bell Polynomials Associated with Random Variables

Abstract

The aim of this paper is to study probabilistic versions of the degenerate Stirling numbers of the second kind and the degenerate Bell polynomials, namely the probabilisitc degenerate Stirling numbers of the second kind associated with \(Y\) and the probabilistic degenerate Bell polynomials associated with \(Y\), which are also degenerate versions of the probabilisitc Stirling numbers of the second and the probabilistic Bell polynomials considered earlier. 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 and recurrence relations for those numbers and polynomials. In addition, we treat the special cases that \(Y\) is the Poisson random variable with parameter \(\alpha (>0)\) and the Bernoulli random variable with probability of success \(p\).

DOI 10.1134/S106192082304009X