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

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.Comment: 6 page

An exotic shuffle relation of ζ({2}m)\zeta(\{2\}^m) and ζ({3,1}n)\zeta(\{3,1\}^n)

In this short note we will provide a new and shorter proof of the following exotic shuffle relation of multiple zeta values: \zeta(\{2\}^m \sha\{3,1\}^n)={2n+m\choose m} \frac{\pi^{4n+2m}}{(2n+1)\cdot (4n+2m+1)!}. This was proved by Zagier when n=0, by Broadhurst when $m=0$, and by Borwein, Bradley, and Broadhurst when m=1. In general this was proved by Bowman and Bradley in \emph{The algebra and combinatorics of shuffles and multiple zeta values}, J. of Combinatorial Theory, Series A, Vol. \textbf{97} (1)(2002), 43--63. Our idea in the general case is to use the method of Borwein et al. to reduce the above general relation to some families of combinatorial identities which can be verified by WZ-method.Comment: 5 page

Duality for Finite Multiple Harmonic q-Series

We define two finite q-analogs of certain multiple harmonic series with an arbitrary number of free parameters, and prove identities for these q-analogs, expressing them in terms of multiply nested sums involving the Gaussian binomial coefficients. Special cases of these identities--for example, with all parameters equal to 1--have occurred in the literature. The special case with only one parameter reduces to an identity for the divisor generating function, which has received some attention in connection with problems in sorting theory. The general case can be viewed as a duality result, reminiscent of the duality relation for the ordinary multiple zeta function.Comment: 12 pages AMSLaTeX. Submitted for publication October 26, 2003; revised September 14, 2004. New title reflects change in emphasis and new section devoted to connections with inverse pairs and Hoffman duality. References added and typos correcte

Signed q-Analogs of Tornheim's Double Series

We introduce signed q-analogs of Tornheim's double series, and evaluate them in terms of double q-Euler sums. As a consequence, we provide explicit evaluations of signed and unsigned Tornheim double series, and correct some mistakes in the literature.Comment: 12 pages, AMSLaTeX. The multinomial notation introduced on page 3 just before Theorem 1 is insufficiently general in version 1, since it may happen that the upper number is negative. This is corrected in version 2, which allows for a negative or even complex upper argumen

Applications of shuffle product to restricted decomposition formulas for multiple zeta values

In this paper we obtain a recursive formula for the shuffle product and apply it to derive two restricted decomposition formulas for multiple zeta values (MZVs). The first formula generalizes the decomposition formula of Euler and is similar to the restricted formula of Eie and Wei for MZVs with one strings of 1's. The second formula generalizes the previous results to the product of two MZVs with one and two strings of 1's respectively.Comment: 11 page