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

Taekyun Kim

 Division of General Education-Mathematics, Kwangwoon University, Seoul 139704, South Korea
Received 13 December 2007; Accepted 29 January 2008

Recommended by Panayiotis D. Siafarikas

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.