Probabilistic Stirling numbers associated with sequences

TL;DR

This paper redefines probabilistic Stirling numbers of both kinds to satisfy orthogonality and inverse relations, explores their degenerate forms, and provides explicit computations, advancing the theoretical understanding of these combinatorial quantities.

Contribution

It introduces a new definition for probabilistic Stirling numbers of the first kind that satisfy key algebraic relations and investigates their degenerate versions.

Findings

Redefinition of probabilistic Stirling numbers of the first kind.

Verification that these numbers satisfy orthogonality and inverse relations.

Explicit computation of the numbers and their degenerate counterparts.

Abstract

Let Y be a random variable whose moment generating function exists in a neighborhood of the origin. Recently, probabilistic Stirling numbers of the first kind and of the second kind associated with Y have been introduced. However, probabilistic stirling number of the first kind based on the cumulant generating function of Y, and probabilistic stirling number of the second kind do not satisfy orthogonality and inverse relations. This paper aims to redefine the probabilistic stirling numbers of the first kind associated with Y such that probabilistic stiirling number of the first kind and probabilistic stirling number of the second kind do satisfy these crucial relations. Furthermore, we investigate their degenerate counterparts, the probabilistic degenerate Stirling numbers of both kinds. We explicitly compute probabilistic stiirling number of the first kind and probabilistic stirling number of the second kind .