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

Abstract

The double zeta function zetaTeX Embedding failed! is a function of two arguments defined by a double Dirichlet series, and was first studied by Euler in response to a letter from Goldbach in 1742. By calculating many examples, Euler inferred a closed form evaluation of the double zeta function in terms of values of the Riemann zeta function in the case when the two arguments are positive integers with opposite parity. Here, we establish a TeX Embedding failed!-analog of Euler's evaluation. That is, we state and
outline the proof of a 1-parameter generalization that reduces to Euler's evaluation in the limit as the parameter q tends to 1. In 2004, the speaker established TeX Embedding failed!-analogs for most of the other formulas satisfied by the multiple zeta function of Euler and Zagier, but at that time a q-analog of the reduction formula for zetaTeX Embedding failed! remained outstanding. Establishing the latter was the result of joint work with Xia Zhou, Dept. of Math., Zhejiang University, Hangzhou, China.

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