Probabilistic degenerate logarithm and heterogeneous stirling numbers
Dae San Kim, Taekyun Kim
TL;DR
The work addresses shortcomings of earlier probabilistic Stirling numbers that lacked orthogonality or did not reduce to classical forms when $Y=1$. It develops a cohesive framework by redefining probabilistic Stirling numbers of the first kind and introducing their degenerate counterparts, alongside probabilistic (degenerate) logarithms associated with a general random variable $Y$. It introduces probabilistic degenerate Daehee and Cauchy numbers, and probabilistic heterogeneous Stirling numbers, plus a probabilistic degenerate Schlömilch identity that mirrors the classical algebraic structure. The results provide a robust bridge between degenerate and probabilistic combinatorics, with explicit identities linking log^Y(1+t) to $S_{2,\lambda}^{Y}(n,k)$ and ensuring the constructs consistently transition to their classical limits, enabling applications in stochastic modeling and umbral calculus.
Abstract
Let Y be a random variable whose moment-generating function exists in some neighborhood of the origin. While probabilistic Stirling numbers of the first and second kind have been introduced, early definitions often failed to satisfy fundamental orthogonality and inverse relations or lacked consistency with classical forms in the case when Y = 1. This paper addresses these limitations by utilizing redefined probabilistic Stirling numbers of the first kind and the second kind alongside their degenerate counterparts. Our primary objective is twofold: first,to introduce the probabilistic (degenerate) logarithm associated with Y, providing explicit expressions for various random variables and defining new probabilistic degenerate Daehee and Cauchy numbers; and second, to investigate probabilistic heterogeneous Stirling numbers and establish a probabilistic degenerate version of the Schlomilch formula, demonstrating that these new frameworks maintain the essential algebraic properties of their classical counterparts.