A probabilistic generalization of the Bell polynomials

Motivated by an interconnection between the probabilistic Stirling numbers of the second kind and the Bell polynomials studied by Adell and Lekuona (in J Number Theory 194:335-355, 2019), we present a probabilistic generalization of the Bell polynomials associated with a random variable Y satisfying suitable moment conditions. We call it probabilistic Bell polynomials. These polynomials are closely related to the probabilistic Stirling numbers of the second kind and generalize the classical Bell polynomials which have various applications in the different disciplines of applied sciences. The exponential generating function and the recurrence relations are obtained. Several convolution identities and some probabilistic extensions of combinatorial sums are also discussed. Interconnections of Poisson, geometric and exponential variates with the probabilistic Bell polynomials and the Stirling numbers of the second kind are studied. A connection to Bernoulli random variate and its application to sum of powers is also obtained. Some specific representations of the probabilistic Stirling numbers of the second kind using discrete exponential and geometric random variates are also derived. Finally, applications to cumulants and Appell polynomials are presented.