Translated Whitney Numbers and Their q-Analogues

Abstract This paper presents natural q-analogues for the translated Whitney numbers. Several combinatorial properties which appear to be q-deformations of those classical ones are obtained. Moreover, we give a combinatorial interpretation of the classical translated Whitney numbers of the first and second kind, and their qanalogues in terms of A-tableaux

Probabilistic degenerate Dowling polynomials associated with random variables

The aim of this paper is to study probabilistic versions of the degenerate Whitney numbers of the second kind and those of the degenerate Dowling polynomials, namely the probabilistic degenerate Whitney numbers of the second kind associated with Y and the probabilistic degenerate Dowling polynomials associated with Y. Here Y is a random variable whose moment generating function exists in some neighborhood of the origin. We derive some properties, explicit expressions, certain identities, recurrence relations and generating functions for those numbers and polynomials. In addition, we investigate their generalizations, namely the probabilistic degenerate r-Whitney numbers of the second kind associated with Y and the probabilistic degenerate r-Dowling polynomials associated with Y, and get similar results to the aforementioned numbers and polynomials.Comment: 14 page

Generalized r-Whitney numbers of the first kind

In this paper, we consider a (p, q)-generalization of the r-Whitney num- ber sequence of the first kind that reduces to it when p = q = 1. We obtain generalizations of some earlier results for the r-Whitney sequence, including recurrence and generating function formulas. We develop a combinatorial interpretation for our generalized numbers in terms of a pair of statistics on the set of r-permutations in which the elements within cycles of a permutation are assigned colors according to certain rules. This allows one to provide combinatorial proofs of various identities, including orthogonality relations. Finally, we consider the (p, q)-Whitney matrix of the first kind and find various factorizations for it

New approach to λ-Stirling numbers

The aim of this paper is to study the $\lambda$-Stirling numbers of both kinds, which are $\lambda$-analogues of Stirling numbers of both kinds. These numbers have nice combinatorial interpretations when $\lambda$ are positive integers. If $\lambda = 1$, then the $\lambda$-Stirling numbers of both kinds reduce to the Stirling numbers of both kinds. We derive new types of generating functions of the $\lambda$-Stirling numbers of both kinds which are related to the reciprocals of the generalized rising factorials. Furthermore, some related identities are also derived from those generating functions. In addition, all the corresponding results to the $\lambda$-Stirling numbers of both kinds are obtained for the $\lambda$-analogues of $r$-Stirling numbers of both kinds, which are generalizations of those numbers