International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 21, Pages 3453-3458

A q-analog of Euler's decomposition formula for the double zeta function

David M. Bradley

Department of Mathematics & Statistics, University of Maine, 5752 Neville Hall, Orono 04469-5752, ME, USA

Received 25 February 2005; Revised 16 September 2005

Copyright © 2005 David M. Bradley. 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.


The double zeta function was first studied by Euler in response to a letter from Goldbach in 1742. One of Euler's results for this function is a decomposition formula, which expresses the product of two values of the Riemann zeta function as a finite sum of double zeta values involving binomial coefficients. Here, we establish a q-analog of Euler's decomposition formula. More specifically, we show that Euler's decomposition formula can be extended to what might be referred to as a “double q-zeta function” in such a way that Euler's formula is recovered in the limit as q tends to 1.