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.


Introduction
The Riemann zeta function is defined for ℜ(s) > 1 by is known as the double zeta function. The sums (2), and more generally those of the form have attracted increasing attention in recent years; see eg. [2,4,5,6,8,9,10,12,13,18]. The survey articles [7,14,22,23,25] provide an extensive list of references. In (3) the sum is over all positive integers k 1 , . . . , k m satisfying the indicated inequalities. Note that with positive integer arguments, s 1 > 1 is necessary and sufficient for convergence.
The problem of evaluating sums of the form (2) for integers s > 1, t > 0 seems to have been first proposed in a letter from Goldbach to Euler [16] in 1742. (See also [15,17] and [1, p. 253].) Among other results for (2), Euler proved that if s − 1 and t − 1 are positive integers, then the decomposition formula holds. A combinatorial proof of Euler's decomposition formula (4) based on the Drinfel'd integral representations [2,4,5,7,8] and the shuffle multiplication rule satisfied by such integrals is given in [5, eq. (10)]. It is of course well-known that (4) can also be proved algebraically by summing the partial fraction decomposition [20, p. 48 over appropriately chosen integers x and c. (See eg. [3].) A q-analog of (3) was independently introduced in [11,21,24] as where Observe that we now have so that (7) represents a generalization of (3). In this note, we establish a q-analog of Euler's decomposition formula (4).

Main Result
Our q-analog of Euler's decomposition formula naturally requires only the m = 1 and m = 2 cases of (7); specifically the q-analogs of (1) and (2) given by We also define, for convenience, the sum We can now state our main result.
Theorem 1. If s − 1 and t − 1 are positive integers, then Observe that the limiting case q = 1 of Theorem 1 reduces to Euler's decomposition formula (4).

A Differential Identity
Our proof of Theorem 1 relies on the following identity. Lemma 1. Let s and t be positive integers, and let x and y be non-zero real numbers. Then for all real q, Proof. Apply the partial differential operator to both sides of the identity Observe that when q = 1, Lemma 1 reduces to the identity 1 from which the partial fraction identity (6) (proved by induction in [19]) trivially follows.

Proof of Theorem 1
First, observe that if s > 1 and t > 1, then from (8), where the inner sum is over all positive integers u and v such that u + v = n. Next, apply Lemma 1 with x = [u] q , y = [v] q , noting that then After interchanging the order of summation, there comes

Final Remarks
In [24], Zhao also gives a formula for the product ζ[s]ζ[t]. However, Zhao's formula is considerably more complicated than ours, as it is derived based on the q-shuffle rule [7,11] satisfied by the Jackson q-integral analogs of the representations (5). Of course, we also have the very simple q-stuffle [11]