r-Bell polynomials in combinatorial Hopf algebras

Abstract

We introduce partial r-Bell polynomials in three combinatorial Hopf algebras. We prove a factorization formula for the generating functions which is a consequence of the Zassenhauss formula. RésuméLes r−polynômes de Bell dans des algèbres de Hopf combinatoires Nous définissons des polynômes r-Bell partiels dans trois algèbres de Hopf combinatoires. Nous prouvons une formule de factorisation pour les fonctions génératrices qui est une conséquence de la formule de Zassenhauss.

Some properties of degenerate complete and partial Bell polynomials

“…. , x n-k+1 ) (see (6), (7)) is the number of partitionings of a set with n elements into k blocks and the coefficient of each monomial is the number of partitioning a set with n elements as the corresponding k blocks.…”

“…The second one is the applications to probability theory which shows certain connections between the modified degenerate complete and partial Bell polynomials and the joint distributions of weighted sums of independent degenerate Poisson random variables. Some of the recent work on Bell polynomials can be found in [1,3,4,6,7,9,10,12,25].…”

An application of the partial r-Bell polynomials on some family of bivariate polynomials

“…Mihoubi et Rahmani [12] introduced and studied these polynomials for which they gave combinatorial and probabilistic interpretations and several properties. Shattuck [19] gave more properties and Chouria and Luque [4] defined three versions of the partial r-Bell polynomials in three combinatorial Hopf algebras. For an application of these polynomials on a family of bivariate polynomials, let us define this family.…”

On a class of polynomials connected to Bell polynomials

“…n+r,k+r ((a i , i ≥ 1) ; (b i , i ≥ 1)) are the partial r-Bell polynomials [5,20,27] defined by n≥k B (r)…”