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

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. In this note, 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.