Commun. Korean Math. Soc. Published online August 27, 2019

Joy Antonette D. Cillar and Roberto B. Corcino
University of San Jose- Recoletos, Cebu Normal University

Abstract : In this paper, we establish an explicit formula for $r$-Dowling numbers in terms of $r$-Whitney Lah and $r$-Whitney numbers of the second kind. This is a generalization of the Qi formula for Bell numbers in terms of Lah and Stirling numbers of the second kind. Moreover, we define the $q,r$-Dowling numbers, $q,r$-Whitney Lah numbers and $q,r$-Whitney numbers of the first kind and obtain several fundamental properties of these numbers such as orthogonality and inverse relations, recurrence relations, and generating functions. Hence, we derive an analogous Qi formula for $q,r$-Dowling numbers expressed in terms of $q,r$-Whitney Lah numbers and $q,r$-Whitney numbers of the second kind.