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# Journal of Combinatorics

## Volume 8 (2017)

### Number 1

### A generalization of the $r$-Whitney numbers of the second kind

Pages: 29 – 55

DOI: http://dx.doi.org/10.4310/JOC.2017.v8.n1.a2

#### Authors

#### Abstract

In this paper, we consider a $(p, q)$-generalization of the $r$-Whitney numbers of the second kind and of the associated $r$-Dowling polynomials. We obtain generalizations of some earlier results for these numbers, including recurrence and generating function formulas, that reduce to them when $p = q = 1$. Furthermore, some of our results appear to be new in the case $p = q = 1$ and thus yield additional formulas for the $r$-Whitney numbers. As a consequence, some new identities are obtained for the q-Stirling and $r$-Whitney numbers. In addition, the log-concavity of our generalized Whitney numbers is shown for certain values of the parameters $p$ and $q$. Finally, we introduce $(p, q)$-Whitney matrices of the second kind and study some of their properties.

#### Keywords

$r$-Whitney number, $r$-Dowling polynomial, $q$-generalization, Whitney matrix

#### 2010 Mathematics Subject Classification

05A15, 05A18, 05A19

Published 2 December 2016