International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 21, Pages 3453-3458
doi:10.1155/IJMMS.2005.3453
A q-analog of Euler's decomposition formula for
the double zeta function
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.
Abstract
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.