Communications of the
Korean Mathematical Society
CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

Article

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Commun. Korean Math. Soc. 2020; 35(1): 21-41

Online first article August 27, 2019      Printed January 31, 2020

https://doi.org/10.4134/CKMS.c180478

Copyright © The Korean Mathematical Society.

A $q$-analogue of Qi formula for $r$-Dowling numbers

Joy Antonette D. Cillar, 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.

Keywords: $r$-Dowling numbers, $r$-Whitney numbers, Lah numbers, recurrence relation, explicit formula, generating function, $q$-analogue

MSC numbers: 05A10, 05A15, 11B65, 11B73