Probabilistic degenerate r-Stirling numbers of the second and probabilistic degenerate r-Bell polynomials

TL;DR

This paper introduces probabilistic extensions of degenerate r-Stirling numbers and Bell polynomials, exploring their properties, explicit formulas, and identities using generating functions, thus broadening the understanding of these combinatorial structures in a probabilistic context.

Contribution

It presents the first probabilistic generalizations of degenerate r-Stirling numbers and Bell polynomials, along with their properties and identities.

Findings

Derived explicit formulas for probabilistic degenerate r-Stirling numbers.

Established generating function representations for the probabilistic polynomials.

Identified key identities and properties linking probabilistic and classical versions.

Abstract

Assume that Y is a random variable whose moment generating function exists in a neighborhood of the origin. We study the probabilistic degenerate r-Stirling numbers of the second kind associated with Y and the probabilistic degenerate r-Bell polynomials associated with Y. They are respectively probabilistic extensions of the degenerate r-Stirling numbers of the second and the degenerate r-Bell polynomials which are studied earlier. The aim of this paper is to obtain their properties, explicit expressions and related identities by means of generating functions.