Journal of Combinatorics

Volume 8 (2017)

Number 1

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

Pages: 29 – 55



Toufik Mansour (Department of Mathematics, University of Haifa, Israel)

José L. Ramírez (Departamento de Matemáticas, Universidad Sergio Arboleda, Bogotá, Colombia)

Mark Shattuck (Department of Mathematics, University of Tennessee, Knoxville, Tenn., U.S.A.)


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.


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

2010 Mathematics Subject Classification

05A15, 05A18, 05A19

Published 2 December 2016