Advances in Difference Equations
Volume 2008 (2008), Article ID 743295, 10 pages
doi:10.1155/2008/743295
Research Article
q-Bernoulli Numbers Associated with q-Stirling Numbers
Division of General Education-Mathematics, Kwangwoon University, Seoul 139704, South Korea
Received 13 December 2007; Accepted 29 January 2008
Academic Editor: Panayiotis D. Siafarikas
Copyright © 2008 Taekyun Kim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We consider Carlitz q-Bernoulli numbers and q-Stirling numbers of the first and the second kinds. From the properties of q-Stirling numbers, we derive many interesting formulas associated with Carlitz q-Bernoulli numbers. Finally, we will prove βn,q=∑m=0n∑k=mn1/(1-q)n+m-k∑d0+⋯+dk=n-kq∑i=0kidis1,q(k,m)(-1)n-m((m+1)/[m+1]q), where βn,q are called Carlitz q-Bernoulli numbers.