A $q$-analogue of the generalized factorial numbers

Seok-Zun Song; Gi-Sang Cheon; Young-Bae Jun; LeRoy B. Beasley

doi:10.4134/JKMS.2010.47.3.645

Journal of the
Korean Mathematical Society JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2010; 47(3): 645-657

Printed May 1, 2010

https://doi.org/10.4134/JKMS.2010.47.3.645

A $q$-analogue of the generalized factorial numbers

Seok-Zun Song, Gi-Sang Cheon, Young-Bae Jun, and LeRoy B. Beasley

Jeju National University, Sungkyunkwan University, Gyeongsang National University, and Utah State University

Abstract

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Abstract

In this paper, more generalized $q$-factorial coefficients are examined by a natural extension of the $q$-factorial on a sequence of any numbers. This immediately leads to the notions of the extended $q$-Stirling numbers of both kinds and the extended $q$-Lah numbers. All results described in this paper may be reduced to well-known results when we set $q=1$ or use special sequences.

Keywords: $q$-factorial, $q$-Stirling numbers, $q$-Lah numbers

MSC numbers: Primary 05A30; Secondary 05A15