Probabilistic degenerate Stirling numbers of the first kind and their applications

TL;DR

This paper introduces probabilistic degenerate Stirling numbers of the first kind linked to a random variable Y, exploring their properties, identities, and applications to normal and gamma distributions.

Contribution

It defines a new degenerate version of probabilistic Stirling numbers of the first kind and investigates their properties and applications, extending previous work by Adell-Benyi.

Findings

Derived properties and identities of the new numbers

Established recurrence relations and explicit formulas

Applied results to normal and gamma distributions

Abstract

Let Y be a random variable whose degenerate moment generating functions exist in some neighborhoods of the origin. The aim of this paper is to study the probabilistic degenerate Stirling numbers of the first kind associated with Y which are constructed from the degenerate cumulant generating function of Y. They are a degenerate version of the probabilistic Stirling numbers of the first kind associated with Y, which were recently introduced by Adell-Benyi. We investigate some properties, related identities, recurrence relations and explicit expressions for those numbers. In addition, we apply our results to the special cases of normal and gamma random variables.