A probabilistic generalization of the Stirling numbers of the second kind

Probabilistic Stirling Numbers of the Second Kind and Applications

“…In general, the iterates in (7) have a cumbersome expression. However, we have the following formulas stated in [3], where it is understood that…”

“…Motivated by various specific problems, different generalizations of the Stirling numbers S( j, m) have been considered in the literature (see, for instance, Hsu and Shiue [11], Luo and Srivastava [13], Cakić et al [6] and El-Desouky et al [9]). In [3], we considered the following probabilistic generalization. Let (Y k ) k≥1 be a sequence of independent copies of a real-valued random variable Y having a finite moment Observe that formula (3) recovers (1) when Y = 1.…”

“…Let (Y k ) k≥1 be a sequence of independent copies of a real-valued random variable Y having a finite moment Observe that formula (3) recovers (1) when Y = 1. The motivations behind definition (3) have to do with certain problems coming from analytic number theory, such as extensions in various ways of the classical formula for sums of powers on arithmetic progressions (cf. [3]) and explicit expressions for higher-order convolutions of Appell polynomials (see [4]).…”

“…Finally, in Sect. 2 we gather some equivalent definitions of S Y ( j, m) when Y is complex-valued without proofs, since they are similar to those previously given in [3] for real-valued random variables Y .…”

Binomial convolution and transformations of Appell polynomials

“…In this section, we collect some definitions and properties, already shown in [12] and [13], which are necessary to state our main results.…”