Probabilistic degenerate logarithm and heterogeneous Stirling numbers

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, 𝑆 𝑌 1 (𝑛, 𝑘) and 𝑆 𝑌 2 (𝑛, 𝑘), 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 Schlömilch formula, demonstrating that these new frameworks maintain the essential algebraic properties of their classical counterparts.