Some properties of degenerate complete and partial Bell polynomials

Abstract

In this paper, we study degenerate complete and partial Bell polynomials and establish some new identities for those polynomials. In addition, we investigate the connections between modified degenerate complete and partial Bell polynomials, which are closely related to the degenerate complete and partial Bell polynomials, and the joint distribution of weighted sums of independent degenerate Poisson random variables.

Concerning Multivariate Bernstein Polynomials and Stochastic Logic

“…Degenerate versions of some special polynomials have been shown to play an important role in various areas. However, not much is known about the properties of these polynomials (cf., e.g, [15][16][17][18][19][20]32] and references thereof). In particular, as a degenerate version of Bernstein polynomials, the degenerate Bernstein polynomials were introduced recently by Kim and Kim in [16].…”

The 2-successive partial Bell polynomials

“…The above polynomials have been considered as important combinatorial tools and have been applied in many different contexts, including: the development of complementary inverse relations for binomial polynomials [9][10][11], the properties of complete and degenerate partial Bell polynomials [7], the establishment of new properties of polynomials such as the degenerate Fubini polynomials [6], and many other topics. This wide application of Bell polynomials inspired the further development of this mathematical tool.…”