Probabilistic multi-Stirling numbers of the second kind associated with random variables

This paper investigates a probabilistic extension of the multi-Stirling numbers of the second kind and a ‘poly' version of the probabilistic degenerate Lah-Bell polynomials. The probabilistic multi-Stirling numbers of the second kind are defined with respect to a random variable satisfying specific moment conditions, generalizing previous work on probabilistic Stirling numbers of the second kind. Explicit expressions and related identities are derived, including special cases where the random variable follows a gamma or Bernoulli distribution. Furthermore, probabilistic degenerate poly-Lah-Bell polynomials are introduced and their explicit expressions are obtained.