Multiple $q$-zeta brackets

The multiple zeta values (MZVs) possess a rich algebraic structure of algebraic relations, which is conjecturally determined by two different (shuffle and stuffle) products of a certain algebra of noncommutative words. In a recent work, Bachmann constructed a $q$-analogue of the MZVs -- the so-called bi-brackets -- for which the two products are dual to each other, in a very natural way. We overview Bachmann's construction and discuss the radial asymptotics of the bi-brackets, its links to the MZVs, and related linear (in)dependence questions of the $q$-analogue.

The goal of this note is to make an algebraic setup for Bachmann's double stuffle relations as well as to demonstrate that those relations indeed reduce to the corresponding stuffle and shuffle relations in the limit as q → 1 − . We also address the reduction of the bi-brackets to the mono-brackets.

Asymptotics
The following result allows one to control the asymptotic behaviour of the bibrackets not only as q → 1 − but also as q approaches radially a root of unity. This produces an explicit version of the asymptotics used in [9] for proving some linear and algebraic results in the case l = 1.
is the generating function of Bernoulli numbers.
Proof. The proof is technical but straightforward.
By moving the constant termλ s to the right-hand side, we get and so on.
Another way to tackle the asymptotic behaviour of the (bi-)brackets is based on the Mellin transform Γ(s) (n 1 d 1 + · · · + n l d l ) s ; see [7,10]. Note that the bijective correspondence between the bi-brackets and the zeta functions can be potentially used for determining the linear relations of the former. A simple illustration is the linear independence of the depth 1 bi-brackets.
Theorem 1. The bi-brackets s 1 r 1 , where 0 ≤ r 1 < s 1 ≤ n, s 1 + r 1 ≤ n, are linearly independent over Q. Therefore, the dimension d BD n of the Q-space spanned by all bi-brackets of weight at most n is bounded from below by ⌊(n + 1) 2 /4⌋ ≥ n(n + 2)/4.
are linearly independent over Q (because of their disjoint sets of poles at s = s 1 and s = r 1 + 1, respectively); thus the corresponding bi-brackets s 1 r 1 are Q-linearly independent as well.
A similar (though more involved) analysis can be applied to describe the Mellin transform of the depth 2 bi-brackets; note that it is more easily done for another q-model we introduce further in Section 3.

The stuffle product
Consider the alphabet Z = {z s,r : s, r = 1, 2, . . . } on the double-indexed letters z s,r of the pre-defined weight s + r − 1. On QZ define the (commutative) product is the generating function of Bernoulli numbers. Note thatλ s = λ s for s ≥ 2, whilê λ 1 = 1 2 = −λ 1 in the notation of Section 1.
As explained in [2] (after the proof of Proposition 2.9), the product ⋄ is also associative. With the help of (4) define the stuffle product on the Q-algebra Q Z recursively by 1 for arbitrary w, v ∈ Q Z and a, b ∈ Z.
Proposition 2. The evaluation map , the latter hence being a Q-algebra as well.
Proof. The proof follows the lines of the proof of [2, Proposition 2.10] based on the identity Modulo the highest weight, the commutative product (4) on Z assumes the form so that the stuffle product (5) reads for arbitrary w, v ∈ Q Z and z s 1 ,r 1 , z s 2 ,r 2 ∈ Z. If we set z s := z s,1 and further restrict the product to the subalgebra Q Z ′ , where Z ′ = {z s : s = 1, 2, . . . }, then Proposition 1 results in the following statement.
, Theorem 2 asserts that the stuffle product (5) of the algebra MD reduces to the stuffle product of the algebra of MZVs in the limit as q → 1 − . This fact has been already established in [2].

The duality
As an alternative extension of the mono-brackets (1) we introduce the multiple q-zeta brackets By applying iteratively the binomial theorem in the forms we see that the Q-algebras spanned by either (3) or (8) coincide. More precisely, the following formulae link the two versions of brackets.
. . . , s l−1 , s l r 1 + r 2 − j 2 , j 2 + r 3 − j 3 , . . . , j l−1 + r l − j l , j l and Z s 1 , . . . , s l r 1 , . . . , r l = Proof. This follows from the rearrangement of the summation indices: Denote by τ the anti-automorphism of the algebra H, interchanging x and y; for example, τ (x 2 yxy) = xyxy 2 . Clearly, τ is an involution preserving both the weight and depth, and it is also an automorphism of the subalgebra H 0 . The duality can be then stated as for any w ∈ H 0 . (9) We also extend τ to Q Z by linearity.
The duality in Proposition 4 is exactly the partition duality given earlier by Bachmann for the model (3).

The dual stuffle product
We can now introduce the product which is dual to the stuffle one. Namely, it is the duality composed with the stuffle product and, again, with the duality: It follows then from Propositions 2 and 4 that , so that it is also a homomorphism of the Q-algebra (Q Z , Note that (7) is also equivalent to the expansion from the right [11,Theorem 9]: The next statement addresses the structure of the dual stuffle product (10) for the words over the sub-alphabet Z ′ = {z s = z s,1 : s = 1, 2, . . . } ⊂ Z. Note that the words from Q Z ′ can be also presented as the words from Q x, xy necessarily ending with xy. Proposition 6. Modulo the highest weight and depth, for arbitrary words w, v ∈ Q + Q x, xy xy and a, b ∈ {x, xy}.
Proof. First note that restricting (11) further modulo the highest depth implies and that we also have The relations already show that for arbitrary words w, v ∈ Q + Q Z and a ′ , b ′ ∈ Z ∪ {y}, where z s 1 ,r 1 . . . z s l−1 ,r l−1 z s l ,r l y = z s 1 ,r 1 . . . z s l−1 ,r l−1 z s l ,r l +1 .
Secondly note that the isomorphism ϕ of Proposition 3 acts trivially on the words from Q Z ′ . Therefore, applying τ ϕ to the both sides of (10) and extracting the homogeneous part of the result corresponding to the highest weight and depth we arrive at y if a = x, xy if a = xy, and using (13) we find out that which implies the desired result.
Theorem 3. For admissible words w = z s 1 . . . z s l and v = z s ′ 1 · · · z s ′ m of weight |w| = s 1 + · · · + s l and |v| = s ′ 1 + · · · + s ′ m , respectively, Proof. Because both ϕ and τ respect the weight, Proposition 6 shows that the only terms that can potentially interfere with the asymptotic behaviour as q → 1 − correspond to the same weight but lower depth. However, according to (10) and (11), the 'shorter' terms do not belong to Q Z ′ , that is, they are linear combinations of the monomials z q 1 ,r 1 . . . z qn,rn with r 1 + · · · + r n = l + m > n, hence r j ≥ 2 for at least one j. The latter circumstance and Proposition 1 then imply lim q→1 − (1 − q) |w|+|v| [z q 1 ,r 1 . . . z qn,rn ] = 0. Theorem 3 asserts that the dual stuffle product (10) restricted from BD to the subalgebra MD reduces to the shuffle product of the algebra of MZVs in the limit as q → 1 − . This result is implicitly stated in [1]. More is true: using (7) and Proposition 6 we obtain Theorem 4. For two words w = z s 1 . . . z s l and v = z s ′ 1 · · · z s ′ m , not necessarily admissible, whenever the MZV on the right-hand side makes sense.
In other words, the q-zeta model of bi-brackets provides us with a (far reaching) regularisation of the MZVs: the former includes the extended double shuffle relations as the limiting q → 1 − case.
Conjecture 1 (Bachmann [1]). The resulting double stuffle (that is, stuffle and dual stuffle) relations exhaust all the relations between the bi-brackets. Equivalently (and simpler), the stuffle relations and the duality exhaust all the relations between the bi-brackets.
We would like to point out that the duality τ from Section 3 also exists for the algebra of MZVs [11,Section 6]. However the two dualities are not at all related: the limiting q → 1 − process squeezes the appearances of x before y in the words x s 1 yx s 2 y . . . x s l y, so that they become x s 1 −1 yx s 2 −1 y . . . x s l −1 y. Furthermore, the duality of MZVs respects the shuffle product: the dual shuffle product coincides with the shuffle product itself. On the other hand, the dual stuffle product of MZVs is very different from the stuffle (and shuffle) products. It may be an interesting problem to understand the double stuffle relations of the algebra of MZVs.

Reduction to mono-brackets
In this final section we present some observations towards another conjecture of Bachmann about the coincidence of the Q-algebras of bi-and mono-brackets. with all s j to be at least 2 (so that there is no appearance of y r with r ≥ 2). The latter statement is already known to be true: Brown [5] proves that one can span the Q-algebra of MZVs by the set with all s j ∈ {2, 3}.
In what follows we analyse the relations for the model (8), because it makes simpler keeping track of the duality relation. We point out from the very beginning that the linear relations given below are all experimentally found (with the check of 500 terms in the corresponding q-expansions) but we believe that it is possible to establish them rigorously using the double stuffle relations given above.
The first presence of the q-zeta brackets that are not reduced to ones from MD by the duality relation happens in weight 3. It is Z 2 2 and we find out that There are 34 totally q-zeta brackets of weight up to 4, , and there is one more relation in this weight between the q-zeta brackets from MD: = 1 + 2x + 4x 2 + 8x 3 + 15x 4 + 28x 5 + 51x 6 + 92x 7 + 165x 8 + 294x 9 + 523x 10 + O(x 11 ).
We can compare this with the count c MD where F n denotes the Fibonacci sequence.
In addition, we would like to point out one more expectation for the algebra of (both mono-and bi-) brackets, which is not shared by other q-models of MZVs: all linear (hence algebraic) relations between them seem to be over Q, not over C(q).

Conjecture 3.
A collection of (bi-)brackets is linearly dependent over C(q) if and only if it is linearly dependent over Q.