Probabilistic Stirling numbers associated with sequences

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 .