# Balanced multiple $\operatorname{q}$-zeta values

Annika Burmester  Faculty of Mathematics, Bielefeld University, Germany.
<EMAIL_ADDRESS>
###### Abstract.

We introduce the balanced multiple q-zeta values. They give a new model for
multiple q-zeta values, whose product formula combines the shuffle and stuffle
product for multiple zeta values in a natural way. Moreover, the balanced
multiple q-zeta values are invariant under a very explicit involution. Thus,
all relations among the balanced multiple q-zeta values are conjecturally of a
very simple shape. Examples of the balanced multiple q-zeta values are the
classical Eisenstein series, and they also contain the combinatorial multiple
Eisenstein series introduced in [BB22]. The construction of the balanced
multiple q-zeta values is done on the level of generating series. We introduce
a general setup relating Hoffman’s quasi-shuffle products to explicit
symmetries among generating series of words, which gives a clarifying approach
to Ecalle’s theory of bimoulds. This allows us to obtain an isomorphism
between the underlying Hopf algebras of words related to the combinatorial bi-
multiple Eisenstein series and the balanced multiple q-zeta values.

###### Key words and phrases:

multiple zeta values, multiple q-zeta values, quasi-shuffle Hopf algebras,
generating series, bimoulds

###### 2020 Mathematics Subject Classification:

11M32, 05A30, 16T05

The author was partially funded by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) – SFB-TRR 358/1 2023 — 491392403.

##  1\. Introduction

In general, a _q-analog_ of some expression is a generalization involving the
variable $q$, which returns the original expression by taking the limit $q\to
1$. For example, a q-analog of some integer $n\geq 1$ is

$\\{n\\}_{q}=\frac{1-q^{n}}{1-q}=1+q+\dots+q^{n-1},$

since $\lim_{q\to 1}\\{n\\}_{q}=n$. In this work we are interested in a
particular kind of q-analogs of multiple zeta values.

### Multiple zeta values.

_Multiple zeta values_ are real numbers defined for integers $k_{1}\geq 2,\
k_{2},\ldots,k_{d}\geq 1$ by

$\zeta(k_{1},\ldots,k_{d})=\sum_{n_{1}>\dots>n_{d}>0}\frac{1}{n_{1}^{k_{1}}\dots
n_{d}^{k_{d}}}.$

We refer to the number $k_{1}+\dots+k_{d}$ as the _weight_ and to the number
$d$ as the _depth_. Multiple zeta values were first studied in depth $2$ by C.
Goldbach and L. Euler more than two centuries ago. During the last three
decades, they became an active field of research. They occur in various areas
of mathematics, such as number theory, algebraic geometry, knot theory,
quantum field theory, and also in mathematical physics. The modern systematic
study of multiple zeta values is initiated in [Hof97], [Za94]. A survey on
achievements in the theory of multiple zeta values can be found in [BGF] and
[Zh20], and all articles related to multiple zeta values are listed in [Hof].
The product of multiple zeta values can be expressed in two different ways,
one is called the _stuffle product_ , and the other one is called the _shuffle
product_. Both products possess a description in terms of quasi-shuffle
algebras (introduced in Section 2). First, consider the alphabet
$\mathcal{Y}=\\{y_{1},y_{2},\ldots\\}$ and let
$\mathbb{Q}\langle\mathcal{Y}\rangle$ be the non-commutative
$\mathbb{Q}$-algebra generated by $\mathcal{Y}$. Then the _stuffle product_
$\ast$ on $\mathbb{Q}\langle\mathcal{Y}\rangle$ is defined by $\mathbf{1}\ast
w=w\ast\mathbf{1}=w$ and

$\displaystyle y_{i}u\ast y_{j}v=y_{i}(u\ast y_{j}v)+y_{j}(y_{i}u\ast
v)+y_{i+j}(u\ast v)$ (1.1)

for all $u,v,w\in\mathbb{Q}\langle\mathcal{Y}\rangle$. The combinatorics of
infinite nested sums imply that there is a surjective algebra morphism

$\displaystyle(\mathbb{Q}\langle\mathcal{Y}\rangle,\ast)$
$\displaystyle\to\mathcal{Z},$ (1.2) $\displaystyle y_{k_{1}}\dots y_{k_{d}}$
$\displaystyle\mapsto\zeta^{\ast}(k_{1},\ldots,k_{d}).$

The elements $\zeta^{\ast}(k_{1},\ldots,k_{d})$ are the _stuffle regularized
multiple zeta values_ , they are uniquely determined by
$\zeta^{\ast}(k_{1},\ldots,k_{d})=\zeta(k_{1},\ldots,k_{d})$ for all
$k_{1}\geq 2,\ k_{2},\ldots,k_{d}\geq 1$ and $\zeta^{\ast}(1)=0$.
Second, consider the finite alphabet $\mathcal{X}=\\{x_{0},x_{1}\\}$ and let
$\mathbb{Q}\langle\mathcal{X}\rangle$ be the non-commutative algebra over
$\mathbb{Q}$ generated by $\mathcal{X}$. The _shuffle product_ on
$\mathbb{Q}\langle\mathcal{X}\rangle$ is recursively defined by
$\mathbf{1}\shuffle w=w\shuffle\mathbf{1}=w$ and

$\displaystyle x_{i}u\shuffle x_{j}v=x_{i}(u\shuffle
x_{j}v)+x_{j}(x_{i}u\shuffle v)$ (1.3)

for all $u,v,w\in\mathbb{Q}\langle\mathcal{X}\rangle$. Define
$\mathfrak{h}^{1}$ to be the subspace of $\mathbb{Q}\langle\mathcal{X}\rangle$
generated by all words not ending in $x_{0}$, i.e., we have
$\mathfrak{h}^{1}=\mathbb{Q}\mathbf{1}+\mathbb{Q}\langle\mathcal{X}\rangle
x_{1}$. Using the iterated integral representation of multiple zeta values one
obtains a surjective algebra morphism

$\displaystyle(\mathfrak{h}^{1},\shuffle)$ $\displaystyle\to\mathcal{Z},$
(1.4) $\displaystyle x_{0}^{k_{1}-1}x_{1}\dots x_{0}^{k_{d}-1}x_{1}$
$\displaystyle\mapsto\zeta^{\shuffle}(k_{1},\ldots,k_{d}).$

The elements $\zeta^{\shuffle}(k_{1},\ldots,k_{d})$ are the _shuffle
regularized multiple zeta values_ , they are uniquely determined by
$\zeta^{\shuffle}(k_{1},\ldots,k_{d})=\zeta(k_{1},\ldots,k_{d})$ for all
$k_{1}\geq 2,\ k_{2},\ldots,k_{d}\geq 1$ and $\zeta^{\shuffle}(1)=0$.
In [IKZ06, Theorem 1] an explicit map is given relating the stuffle and the
shuffle regularized multiple zeta values, which allows comparing the two
product formulas given in (1.2) and (1.4). This yields the _extended double
shuffle relations_ among multiple zeta values, which are conjectured to give
all algebraic relations in $\mathcal{Z}$ ([IKZ06]).
Quasi-shuffle products can be lifted to symmetries among generating series
(Section 3). Consider the generating series of the shuffle resp. stuffle
regularized multiple zeta values

$\displaystyle\mathfrak{z}_{d}^{\bullet}(X_{1},\ldots,X_{d})$
$\displaystyle=\sum_{k_{1},\ldots,k_{d}\geq
1}\zeta^{\bullet}(k_{1},\ldots,k_{d})X_{1}^{k_{1}-1}\dots
X_{d}^{k_{d}-1}\in\mathcal{Z}\llbracket X_{1},\ldots,X_{d}\rrbracket,\qquad
d\geq 1,$

with $\bullet\in\\{\ast,\shuffle\\}$. The stuffle and shuffle product in depth
$2$ translate to

$\displaystyle\mathfrak{z}_{1}^{\ast}(X_{1})\mathfrak{z}_{1}^{\ast}(X_{2})$
$\displaystyle=\mathfrak{z}_{2}^{\ast}(X_{1},X_{2})+\mathfrak{z}_{2}^{\ast}(X_{2},X_{1})+\frac{\mathfrak{z}_{1}^{\ast}(X_{1})-\mathfrak{z}_{1}^{\ast}(X_{2})}{X_{1}-X_{2}},$
(1.5)
$\displaystyle\mathfrak{z}_{1}^{\shuffle}(X_{1})\mathfrak{z}_{1}^{\shuffle}(X_{2})$
$\displaystyle=\mathfrak{z}_{2}^{\shuffle}(X_{1}+X_{2},X_{2})+\mathfrak{z}_{2}^{\shuffle}(X_{1}+X_{2},X_{1}).$
(1.6)

In general depths, there are recursive formulas for the stuffle and shuffle
product in terms of generating series of words ([Ih07, Proposition 8]). Those
product formulas explain Ecalle’s notion of _symmetril_ and _symmetral_ moulds
([Ec02, equation (6), (8)]).

### Multiple q-zeta values.

For a better understanding of the structure of the multiple zeta values, we
study particular q-analogs of them. Including the additional variable $q$
reveals new structures. For example, one obtains an involution relating the
two product formulas of multiple zeta values, or a derivation with respect to
$q$ known from the theory of quasi-modular forms.
We use the model-free approach to q-analogs of multiple zeta values given in
[BK20]. To integers $s_{1}\geq 1,\ s_{2},\ldots,s_{l}\geq 0$ and polynomials
$R_{1}\in t\mathbb{Q}[t],\ R_{2},\ldots,R_{l}\in$ $\mathbb{Q}[t]$, associate
the _generic multiple q-zeta value_

$\zeta_{q}(s_{1},...,s_{l};R_{1},...,R_{l})=\sum_{n_{1}>\dots>n_{l}>0}\frac{R_{1}(q^{n_{1}})}{(1-q^{n_{1}})^{s_{1}}}\cdots\frac{R_{l}(q^{n_{l}})}{(1-q^{n_{l}})^{s_{l}}}\in\mathbb{Q}\llbracket
q\rrbracket.$

For $s_{1}\geq 2$, $s_{2},\ldots,s_{l}\geq 1$, a generic multiple q-zeta
values $\zeta_{q}(s_{1},\ldots,s_{l};R_{1},\ldots,R_{l})$ is indeed a
(modified) q-analog of multiple zeta values (Proposition 8.2). The product of
any two generic multiple q-zeta values is a $\mathbb{Q}$-linear combination of
generic multiple q-zeta values.

######  Definition.

The _algebra of multiple q-zeta values_ is the subalgebra of
$\mathbb{Q}\llbracket q\rrbracket$ given by

$\mathcal{Z}_{q}=\operatorname{span}_{\mathbb{Q}}\\!\big{\\{}\zeta_{q}(s_{1},...,s_{l};R_{1},...,R_{l})\
\big{|}\ l\geq 0,s_{1}\geq 1,\ s_{2},...,s_{l}\geq 0,\deg(R_{j})\leq
s_{j}\big{\\}},$

where we set $\zeta_{q}(\emptyset;\emptyset)=1$.

The additional requirement on the degree of the polynomials is natural. For
example, $\mathcal{Z}_{q}$ is related to polynomial functions on partitions
([Ba19], [BI22]), which implies the existence of spanning sets of
$\mathcal{Z}_{q}$ invariant under some involution. The space $\mathcal{Z}_{q}$
also occurs in enumerative geometry. More precisely, A. Okounkov conjectured
that certain generating series of Chern numbers on Hilbert schemes of points
are always contained in the space $\mathcal{Z}_{q}$ ([Ok14]).
A _model_ of multiple q-zeta values is a particular assumption on the
polynomials $R_{i}$ (usually these particular polynomials form a basis of
$\mathbb{Q}[t]$) or just a spanning set of $\mathcal{Z}_{q}$. About twenty
years ago the first models for multiple q-zeta values were introduced
independently by D. Bradley, K. Schlesinger, J. Zhao, and W. Zudilin ([Bra05],
[Sc01], [Zh07], [Zu03]). Since then several more models appeared in the
literature, all of them are contained in the above defined space
$\mathcal{Z}_{q}$. An overview on the various models of multiple q-zeta values
and their relations is given in [BK20] and [Bri21].
Traditionally, the models of multiple q-zeta values focus on a q-analog of the
shuffle product given in (1.2) or the stuffle product given in (1.4) for
multiple zeta values but do not combine them. So, usually it is difficult to
describe a q-analog of the extended double shuffle relations. In rather recent
articles on multiple q-zeta values by H. Bachmann ([Ba19]) and by K. Ebrahimi-
Fard, D. Manchon, and J. Singer ([EMS16]) the focus changed to obtaining a
product formula and some invariance under an involution. Combining the product
formula and the involution one easily derives a q-analog of the extended
double shuffle relations. In joint work with H. Bachmann ([BB22]), we
constructed the combinatorial bi-multiple Eisenstein series, which satisfy a
weight-graded product formula and are invariant under some weight-homogeneous
involution.

### Balanced multiple q-zeta values.

A main result of this article gives a new model of multiple q-zeta values,
which also satisfies weight-homogeneous relations and is closely related to
the Schlesinger-Zudilin multiple q-zeta values studied in [EMS16]. We call the
elements in this model the _balanced multiple q-zeta values_
$\zeta_{q}(s_{1},\ldots,s_{l}),\ s_{1}\geq 1,\ s_{2},\ldots,s_{l}\geq 0$
(Definition 10.1). The classical Eisenstein series and their derivatives are
examples of balanced multiple q-zeta values. Moreover, the balanced multiple
q-zeta values can be used to conjecturally describe all relations among
multiple Eisenstein series. This new model also gives an explicit description
of a conjectural weight-grading on the algebra $\mathcal{Z}_{q}$, which
extends the weight-grading of the algebra
$\widetilde{\mathcal{M}}^{\mathbb{Q}}(\operatorname{SL}_{2}(\mathbb{Z}))$ of
quasi-modular forms with rational coefficients (introduced in [KZ95]). The
balanced multiple q-zeta values are q-analogs of multiple zeta values, taking
the limit $q\to 1$ always yields an element in the algebra of multiple zeta
values (Proposition 11.5, Remark 11.6). An advantage of the balanced multiple
q-zeta values is that they satisfy a product formula, which can be seen as a
balanced combination of the shuffle and stuffle product for the multiple zeta
values (Theorem 10.4). Therefore, they provide a setup to study both products
of the multiple zeta values at the same time. Moreover, the balanced multiple
q-zeta values satisfy linear relations coming from a particular simple
involution. They also possess a description in terms of a finite alphabet
similar to the case of multiple zeta values (Theorem 11.4).
Consider the alphabet

$\mathcal{B}=\\{b_{0},b_{1},b_{2},\ldots\\}$

and denote by $\mathbb{Q}\langle\mathcal{B}\rangle$ the free non-commutative
$\mathbb{Q}$-algebra generated by $\mathcal{B}$. The _balanced quasi-shuffle
product_ $\ast_{b}$ is the quasi-shuffle product on
$\mathbb{Q}\langle\mathcal{B}\rangle$ recursively defined by
$\mathbf{1}\ast_{b}w=w\ast_{b}\mathbf{1}=w$ and

$\displaystyle
b_{i}u\ast_{b}b_{j}v=b_{i}(u\ast_{b}b_{j}v)+b_{j}(b_{i}u\ast_{b}v)+\begin{cases}b_{i+j}(u\ast_{b}v)&\text{
if }i,j\geq 1,\\\ 0&\text{ else}\end{cases}$ (1.7)

for all $b_{i},b_{j}\in\mathcal{B},\
u,v,w\in\mathbb{Q}\langle\mathcal{B}\rangle$. Restricted to the letters
$b_{i}$ for $i\geq 1$ the product $\ast_{b}$ is the stuffle product given in
(1.1) and modulo words containing the letters $b_{i}$ for $i\geq 2$ we recover
the shuffle product given in (1.3), therefore we call $\ast_{b}$ the balanced
quasi-shuffle product. The product $\ast_{b}$ is exactly the associated
weight-graded product to the quasi-stuffle product of the Schlesinger-Zudilin
multiple q-zeta values ([Si15, Theorem 3.3]).
Denote by $\mathbb{Q}\langle\mathcal{B}\rangle^{0}$ the subalgebra of
$\mathbb{Q}\langle\mathcal{B}\rangle$ generated by all words which do not
start in $b_{0}$, i.e., we have
$\mathbb{Q}\langle\mathcal{B}\rangle^{0}=\mathbb{Q}\langle\mathcal{B}\rangle\backslash
b_{0}\mathbb{Q}\langle\mathcal{B}\rangle$. Let
$\tau:\mathbb{Q}\langle\mathcal{B}\rangle^{0}\to\mathbb{Q}\langle\mathcal{B}\rangle^{0}$
be the involution given by $\tau(\mathbf{1})=\mathbf{1}$ and

$\displaystyle\tau(b_{k_{1}}b_{0}^{m_{1}}\dots
b_{k_{d}}b_{0}^{m_{d}})=b_{m_{d}+1}b_{0}^{k_{d}-1}\dots
b_{m_{1}+1}b_{0}^{k_{1}-1}.$

A main result of this work is the following.

######  Theorem (Theorem 10.4).

There is a $\tau$-invariant, surjective algebra morphism

$\displaystyle(\mathbb{Q}\langle\mathcal{B}\rangle^{0},\ast_{b})$
$\displaystyle\to\mathcal{Z}_{q},$ $\displaystyle b_{s_{1}}\dots b_{s_{l}}$
$\displaystyle\mapsto\zeta_{q}(s_{1},\ldots,s_{l}),$

where $\zeta_{q}(s_{1},\ldots,s_{l})$ are the balanced multiple q-zeta values
introduced in Definition 10.1.

To the balanced multiple q-zeta value $\zeta_{q}(s_{1},\ldots,s_{l})$
associate the _weight_

$s_{1}+\dots+s_{l}+\\#\\{i\mid s_{i}=0\\}.$

The balanced quasi-shuffle product as well as the $\tau$-invariance of the
balanced multiple q-zeta values are homogeneous for the weight. Since we
expect that all algebraic relations in $\mathcal{Z}_{q}$ can be deduced from
these to sets of relations among balanced multiple q-zeta values (Conjecture
10.5), we expect the algebra $\mathcal{Z}_{q}$ to be graded with respect to
this weight.
Essential for the construction of the balanced multiple q-zeta values are the
_combinatorial bi-multiple Eisenstein series_ introduced in [BB22] (Definition
9.7), which also possess a description in terms of quasi-shuffle algebras.
Consider the alphabet $\mathcal{Y}^{\operatorname{bi}}=\\{y_{k,m}\mid k\geq
1,m\geq 0\\}$ and let
$\mathbb{Q}\langle\mathcal{Y}^{\operatorname{bi}}\rangle$ be the non-
commutative $\mathbb{Q}$-algebra generated by
$\mathcal{Y}^{\operatorname{bi}}$. The _stuffle product_ $\ast$ on
$\mathbb{Q}\langle\mathcal{Y}^{\operatorname{bi}}\rangle$ is recursively
defined by $\mathbf{1}\ast w=w\ast\mathbf{1}=w$ and

$y_{k_{1},m_{2}}u\ast y_{k_{2},m_{2}}v=y_{k_{1},m_{1}}(u\ast
y_{k_{2},m_{2}}v)+y_{k_{2},m_{2}}(y_{k_{1},m_{1}}u\ast
v)+y_{k_{1}+k_{2},m_{1}+m_{2}}(u\ast v)$

for all $u,v,w\in\mathbb{Q}\langle\mathcal{Y}^{\operatorname{bi}}\rangle$.
This is the canonical bi-version of the stuffle product given in (1.1).
Moreover, there is an involution
$\operatorname{swap}:\mathbb{Q}\langle\mathcal{Y}^{\operatorname{bi}}\rangle\to\mathbb{Q}\langle\mathcal{Y}^{\operatorname{bi}}\rangle$
(Definition 7.9) closely related to conjugation of partitions.
Following the general explanations in Section 3, we will express the stuffle
product $\ast$ on $\mathbb{Q}\langle\mathcal{Y}^{\operatorname{bi}}\rangle$
and the balanced quasi-shuffle product $\ast_{b}$ on
$\mathbb{Q}\langle\mathcal{B}\rangle^{0}$ by a recursive formula on generating
series of words (Theorem 4.1, 5.1). Moreover, we will obtain a _regularization
map_

$\operatorname{reg}:(\mathbb{Q}\langle\mathcal{B}\rangle,\ast_{b})\to(\mathbb{Q}\langle\mathcal{B}\rangle^{0},\ast_{b}),$

which also possesses a description on generating series of words (Theorem
6.3). This regularization allows defining a _regularized coproduct_
$\Delta_{\operatorname{dec}}^{0}:\mathbb{Q}\langle\mathcal{B}\rangle^{0}\to\mathbb{Q}\langle\mathcal{B}\rangle^{0}\otimes\mathbb{Q}\langle\mathcal{B}\rangle^{0}$
compatible with the balanced quasi-shuffle product. The lift to generating
series of words allows us to show the following.

######  Theorem (Theorem 7.10).

There is an isomorphism of weight-graded Hopf algebras

$\displaystyle\varphi_{\\#}:(\mathbb{Q}\langle\mathcal{Y}^{\operatorname{bi}}\rangle,\ast,\Delta_{\operatorname{dec}})\to(\mathbb{Q}\langle\mathcal{B}\rangle^{0},\ast_{b},\Delta_{\operatorname{dec}}^{0}),$

which satisfies
$\varphi_{\\#}\circ\operatorname{swap}=\tau\circ\varphi_{\\#}$.

The stuffle product $\ast$ as well as the involution $\operatorname{swap}$ on
$\mathbb{Q}\langle\mathcal{Y}^{\operatorname{bi}}\rangle$ can be rephrased in
terms of bimoulds (compare to Section 3), this leads to the symmetries usually
called _symmetril_ and _swap invariant_ (those symmetries are extensively
studied in [Ec11], [Sc15]). An example of a symmetril and swap invariant
bimould is the bimould $\mathfrak{G}=(\mathfrak{G}_{d})_{d\geq 0}$ of
generating series of the combinatorial bi-multiple Eisenstein series
(Definition 9.5). On the other hand, the balanced quasi-shuffle product
$\ast_{b}$ and the involution on $\mathbb{Q}\langle\mathcal{B}\rangle^{0}$
also lead to symmetries of bimoulds, which we will refer to as _b-symmetril_
and _$\tau$ -invariant_. Theorem 7.10 gives an isomorphism between symmetril
and swap invariant bimoulds and b-symmetril and $\tau$-invariant bimoulds.
Applying this isomorphism to the bimould
$\mathfrak{G}=(\mathfrak{G}_{d})_{d\geq 0}$ of generating series of the
combinatorial bi-multiple Eisenstein series yields the bimould
$\mathfrak{B}=(\mathfrak{B}_{d})_{d\geq 0}$ of generating series of the
balanced multiple q-zeta values (Definition 10.1). For example, the
b-symmetrility of $\mathfrak{B}$ in depth $2$ reads
$\displaystyle\mathfrak{B}_{1}\binom{X_{1}}{Y_{1}}\cdot\mathfrak{B}_{1}\binom{X_{2}}{Y_{2}}=\mathfrak{B}_{2}\binom{X_{2},X_{1}}{Y_{2},Y_{1}+Y_{2}}+\mathfrak{B}_{2}\binom{X_{1},X_{2}}{Y_{1},Y_{1}+Y_{2}}+\frac{\mathfrak{B}_{1}\binom{X_{1}}{Y_{1}+Y_{2}}-\mathfrak{B}_{1}\binom{X_{2}}{Y_{1}+Y_{2}}}{X_{1}-X_{2}}.$
It combines the product formulas on the generating series of the stuffle and
shuffle regularized multiple zeta values given in (1.5). The results in
Section 7 for the generating series of words allow us to deduce the previously
presented properties of the balanced multiple q-zeta values from the
b-symmetrility and the $\tau$-invariance of $\mathfrak{B}$.

### Structure of the paper.

We start by recalling Hoffman’s quasi-shuffle Hopf algebras ([Hof00]) in
Section 2. Then in Section 3 we develop a general approach relating quasi-
shuffle products to symmetries among generating series resp. bimoulds. These
general observations will be applied to the quasi-shuffle algebras
$(\mathbb{Q}\langle\mathcal{Y}^{\operatorname{bi}}\rangle,\ast)$ and
$(\mathbb{Q}\langle\mathcal{B}\rangle^{0},\ast_{b})$ in Section 4 and 5. In
Section 6, we will introduce the regularization and the regularized coproduct
for $(\mathbb{Q}\langle\mathcal{B}\rangle^{0},\ast_{b})$ and reformulate this
in terms of generating series. This allows to obtain the Hopf algebra
isomorphism
$(\mathbb{Q}\langle\mathcal{Y}^{\operatorname{bi}}\rangle,\ast,\Delta_{\operatorname{dec}})\to(\mathbb{Q}\langle\mathcal{B}\rangle^{0},\ast_{b},\Delta_{\operatorname{dec}}^{0})$
in Section 7. In Section 8 we recall the definition of the algebra
$\mathcal{Z}_{q}$ of multiple q-zeta values (as given in [BK20]) and explain
shortly its relations to multiple zeta values, quasi-modular forms, and
partitions. Then in Section 9 we introduce the combinatorial bi-multiple
Eisenstein series constructed in [BB22], which form a spanning set of
$\mathcal{Z}_{q}$. Applying the results in Section 7 to the combinatorial bi-
multiple Eisenstein series, we will obtain the balanced multiple q-zeta values
and some of its most important properties in Section 10. Finally, we give some
more properties of the balanced multiple q-zeta values in Section 11.

### Acknowledgment.

This work was mainly part of my PhD thesis. Therefore, I deeply thank my
supervisor Ulf Kühn for many helpful comments and discussions on these
contents. Moreover, I would like to thank Claudia Alfes-Neumann, Henrik
Bachmann, Jan-Willem van Ittersum, and Koji Tasaka for valuable comments on
these topics within my PhD project or on earlier versions of this paper.

##  2\. Quasi-shuffle Hopf algebras

We give a short introduction to quasi-shuffle Hopf algebras, which were
introduced and studied in [Hof00] and [HI17]. They will be used later to
describe the product of the algebra of multiple q-zeta values in terms of
particular spanning sets.
In the following, $R$ is some arbitrary fixed $\mathbb{Q}$-algebra. Let
$\mathcal{A}$ be an _alphabet_ , this means $\mathcal{A}$ is a countable set
whose elements are called _letters_. By $R\mathcal{A}$ denote the $R$-module
spanned by the letters of $\mathcal{A}$ and let $R\langle\mathcal{A}\rangle$
be the free non-commutative algebra generated by the alphabet $\mathcal{A}$.
The monic monomials in $R\langle\mathcal{A}\rangle$ are called _words_ with
letters in $\mathcal{A}$, and the set of all words is denoted by
$\mathcal{A}^{*}$. Moreover, let $\mathbf{1}$ be the empty word.

######  Definition 2.1.

Let $\diamond:R\mathcal{A}\times R\mathcal{A}\to R\mathcal{A}$ be an
associative and commutative product. Define the _quasi-shuffle product_
$\ast_{\diamond}$ on $R\langle\mathcal{A}\rangle$ recursively by
$\mathbf{1}\ast_{\diamond}w=w\ast_{\diamond}\mathbf{1}=w$ and

$\displaystyle
au\ast_{\diamond}bv=a(u\ast_{\diamond}bv)+b(au\ast_{\diamond}v)+(a\diamond
b)(u\ast_{\diamond}v)$

for all $u,v,w\in R\langle\mathcal{A}\rangle$ and $a,b\in\mathcal{A}$.

Note that a quasi-shuffle product $\ast_{\diamond}$ can be equally defined
recursively from the left and from the right.

######  Example 2.2.

1\. Define

$a\diamond b=0\text{ for all }a,b\in\mathcal{A},$

then we get the well-known _shuffle product_ , which is usually denoted by
$\shuffle$.
2\. Consider the bi-alphabet $\mathcal{Y}^{\operatorname{bi}}=\\{y_{k,m}\mid
k\geq 1,\ m\geq 0\\}$ and on $R\mathcal{Y}^{\operatorname{bi}}$ define the
product

$\displaystyle y_{k_{1},m_{1}}\diamond
y_{k_{2},m_{2}}=y_{k_{1}+k_{2},m_{1}+m_{2}}.$

The obtained quasi-shuffle product is called the _stuffle product_ and is
denoted by $\ast$. It appears in the context of the combinatorial bi-multiple
Eisenstein series (Corollary 9.10) and is exactly the associated weight-graded
product to the quasi-shuffle product of Bachmann’s bi-brackets ([Ba19, Theorem
3.6]).
3\. Consider the alphabet $\mathcal{B}=\\{b_{0},b_{1},b_{2},\ldots\\}$ and
define on $R\mathcal{B}$ the product

$b_{i}\diamond_{b}b_{j}=\begin{cases}b_{i+j},&\text{if }i,j\geq 1,\\\
0&\text{else}.\end{cases}$

This quasi-shuffle product occurs for balanced multiple q-zeta values (Theorem
10.4) and will be called the _balanced quasi-shuffle product_ , it is denoted
by $\ast_{b}$. It is exactly the associated weight-graded product to the
quasi-shuffle product of the Schlesinger-Zudilin multiple q-zeta values
([Si15, Theorem 3.3]).

Let
$\Delta_{\operatorname{dec}}:\mathbb{Q}\langle\mathcal{A}\rangle\to\mathbb{Q}\langle\mathcal{A}\rangle\otimes\mathbb{Q}\langle\mathcal{A}\rangle$
be the _deconcatenation product_ , i.e., we have

$\displaystyle\Delta_{\operatorname{dec}}(w)=\sum_{uv=w}u\otimes v\quad\text{
for each word }w\in\mathbb{Q}\langle\mathcal{A}\rangle.$ (2.1)

We close this section with the following structure theorem of Hoffman.

######  Theorem 2.3.

([Hof00, Theorem 3.2, 3.3]) The tuple
$(R\langle\mathcal{A}\rangle,\ast_{\diamond},\Delta_{\operatorname{dec}})$ is
an associative, commutative Hopf algebra. There is an isomorphism between all
quasi-shuffle Hopf algebras defined over the same alphabet given by some
exponential map. ∎

It is well-known that the shuffle algebra
$(\mathbb{Q}\langle\mathcal{A}\rangle,\shuffle)$ is a free polynomial algebra
on the Lyndon words ([Re93, Theorem 4.9 (ii)]). The theorem shows that this
holds for any quasi-shuffle algebra.

##  3\. Quasi-shuffle products and generating series

Inspired by [Ih07, Section 7], we will explain how quasi-shuffle products can
be translated into symmetries of commutative generating series resp. bimoulds.
This gives a connection between Hoffman’s theory of quasi-shuffle Hopf
algebras ([Hof00], [HI17]) and Ecalle’s theory of bimoulds ([Ec11], [Sc15]).
We will use this general setup later to construct a particular nice-behaving
model for multiple q-zeta values and explain its properties.
Consider some given quasi-shuffle algebra
$(\mathbb{Q}\langle\mathcal{A}\rangle,\ast_{\diamond})$. Define the _generic
diagonal series_ of $\mathbb{Q}\langle\mathcal{A}\rangle$ by

$\displaystyle\mathcal{W}(\mathcal{A})=\sum_{w\in\mathcal{A}^{*}}w\otimes w.$

We want to apply $\mathbb{Q}$-linear maps to the first factors, usually
denoted by $\varphi$, or to the second factors, usually denoted by $\rho$, of
$\mathcal{W}(\mathcal{A})$ to get generating series of different kinds and
describe the resulting properties. We begin with an abstract discussion and
later a more detailed explanation of special cases is given.
Let $\operatorname{dep}:\mathcal{A}^{*}\to\mathbb{Z}_{\geq 0}$ be a _depth
map_ compatible with concatenation, i.e., we have

$\operatorname{dep}(uv)=\operatorname{dep}(u)+\operatorname{dep}(v),\qquad
u,v\in\mathcal{A}^{*}.$

Denote by $(\mathcal{A}^{*})^{(d)}$ the set of all words in $\mathcal{A}^{*}$
of depth $d$ and by $\mathbb{Q}\langle\mathcal{A}\rangle^{(d)}$ the space
spanned by $(\mathcal{A}^{*})^{(d)}$.

######  Definition 3.1.

Let
$\rho_{\mathcal{A}}:\mathbb{Q}\langle\mathcal{A}\rangle\to\mathbb{Q}[Z_{1},Z_{2},\ldots]$
be a $\mathbb{Q}$-linear map having the following properties with respect to
the depth map:

  * (i)

There is a strictly increasing sequence $\ell(0)<\ell(1)<\ell(2)<\dots$ of
non-negative integers, such that for each $d\geq 0$ the restriction of
$\rho_{\mathcal{A}}$ to $\mathbb{Q}\langle\mathcal{A}\rangle^{(d)}$ is an
injective $\mathbb{Q}$-linear map

$\rho_{\mathcal{A}}|_{\mathbb{Q}\langle\mathcal{A}\rangle^{(d)}}:\mathbb{Q}\langle\mathcal{A}\rangle^{(d)}\to\mathbb{Q}[Z_{1},\ldots,Z_{\ell(d)}].$

  * (ii)

We have

$\rho_{\mathcal{A}}(uv)=\rho_{\mathcal{A}}(u)\rho_{\mathcal{A}}^{[\ell(n)]}(v),\qquad
u\in\mathbb{Q}\langle\mathcal{A}\rangle^{(n)},\
v\in\mathbb{Q}\langle\mathcal{A}\rangle,$

where $\rho_{\mathcal{A}}^{[n]}$ denotes the $\mathbb{Q}$-linear map obtained
from $\rho_{\mathcal{A}}$ by shifting the variables $Z_{i}$ to $Z_{n+i}$, so
$\rho_{\mathcal{A}}^{[n]}\Big{(}\mathbb{Q}\langle\mathcal{A}\rangle^{(d)}\Big{)}\subset\mathbb{Q}[Z_{n+1},\ldots,Z_{n+\ell(d)}]$.

######  Definition 3.2.

Let
$\rho_{\mathcal{A}}:\mathbb{Q}\langle\mathcal{A}\rangle\to\mathbb{Q}[Z_{1},Z_{2},\ldots]$
be a $\mathbb{Q}$-linear map as in Definition 3.1. The (commutative)
_generating series of words_ in $\mathbb{Q}\langle\mathcal{A}\rangle$
associated to $\rho_{\mathcal{A}}$ are given by

$\rho_{\mathcal{A}}(\mathcal{W})_{d}(Z_{1},\ldots,Z_{\ell(d)})=\sum_{w\in(\mathcal{A}^{*})^{(d)}}w\rho(w)\in\mathbb{Q}\langle\mathcal{A}\rangle\llbracket
Z_{1},\ldots,Z_{\ell(d)}\rrbracket,\qquad d\geq 0.$

We will often drop the index $d$ and simply write
$\rho_{\mathcal{A}}(\mathcal{W})(Z_{1},\ldots,Z_{\ell(d)})$.

######  Example 3.3.

Consider the alphabet $\mathcal{Y}=\\{y_{1},y_{2},y_{3},\ldots\\}$ and define
the depth of a word in $\mathbb{Q}\langle\mathcal{Y}\rangle$ by

$\operatorname{dep}(y_{k_{1}}\dots y_{k_{d}})=d.$

The map

$\displaystyle\rho_{\mathcal{Y}}:\mathbb{Q}\langle\mathcal{Y}\rangle$
$\displaystyle\to\mathbb{Q}[Z_{1},Z_{2},\ldots],$ $\displaystyle
y_{k_{1}}\dots y_{k_{d}}$ $\displaystyle\mapsto Z_{1}^{k_{1}-1}\dots
Z_{d}^{k_{d}-1}$

is a $\mathbb{Q}$-linear map satisfying the conditions in Definition 3.1 with
$\ell(d)=d$. The associated generating series of words are given by
$\rho_{\mathcal{Y}}(\mathcal{W})_{0}=\mathbf{1}$ and

$\rho_{\mathcal{Y}}(\mathcal{W})_{d}(Z_{1},\ldots,Z_{d})=\sum_{k_{1},\ldots,k_{d}\geq
1}y_{k_{1}}\dots y_{k_{d}}Z_{1}^{k_{1}-1}\dots Z_{d}^{k_{d}-1},\qquad d\geq
1.$

In the following sections, we will compute more involved examples.

######  Proposition 3.4.

Let $\rho_{\mathcal{A}}:\mathbb{Q}\langle
A\rangle\to\mathbb{Q}[Z_{1},Z_{2},\ldots]$ be a $\mathbb{Q}$-linear map as in
Definition 3.1 with $\ell(d_{1})+\ell(d_{2})=\ell(d_{1}+d_{2})$ for all
$d_{1},d_{2}\geq 0$. Then the $\mathbb{Q}\llbracket
Z_{1},Z_{2},\ldots\rrbracket$-linear extension of the concatenation product
$\cdot$ satisfies

$\rho_{\mathcal{A}}(\mathcal{W})_{n}(Z_{1},\ldots,Z_{\ell(n)})\cdot\rho_{\mathcal{A}}(\mathcal{W})_{d-n}(Z_{\ell(n)+1},\ldots,Z_{\ell(d)})=\rho_{\mathcal{A}}(\mathcal{W})_{d}(Z_{1},\ldots,Z_{\ell(d)})$

for all $0\leq n\leq d$.

###### Proof.

For $n=0,d$ the formula is obvious. For $0<n<d$, compute

$\displaystyle\rho_{\mathcal{A}}(\mathcal{W})_{n}(Z_{1},\ldots,Z_{\ell(n)})\cdot\rho_{\mathcal{A}}(\mathcal{W})_{d-n}(Z_{\ell(n)+1},\ldots,Z_{\ell(d)})=\sum_{u\in(\mathcal{A}^{*})^{(n)}}\sum_{v\in(\mathcal{A}^{*})^{(d-n)}}uv\rho_{\mathcal{A}}(u)\rho_{\mathcal{A}}^{[\ell(n)]}(v)$
$\displaystyle=\sum_{u\in(\mathcal{A}^{*})^{(n)}}\sum_{v\in(\mathcal{A}^{*})^{(d-n)}}uv\rho_{\mathcal{A}}(uv)=\sum_{w\in(\mathcal{A}^{*})^{(d)}}w\rho_{\mathcal{A}}(w)=\rho_{\mathcal{A}}(\mathcal{W})_{d}(Z_{1},\ldots,Z_{\ell(d)}).$

∎

If we extend the quasi-shuffle product $\ast_{\diamond}$ on
$\mathbb{Q}\langle\mathcal{A}\rangle$ to
$\mathbb{Q}\langle\mathcal{A}\rangle\llbracket Z_{1},Z_{2},\ldots\rrbracket$
by $\mathbb{Q}\llbracket Z_{1},Z_{2},\ldots\rrbracket$-linearity, then by
definition of the quasi-shuffle product $\ast_{\diamond}$ we have for all
$0\leq n\leq d$ that

$\rho_{\mathcal{A}}(\mathcal{W})_{n}(Z_{1},\ldots,Z_{\ell(n)})\ast_{\diamond}\rho_{\mathcal{A}}(\mathcal{W})_{d-n}(Z_{\ell(n)+1},\ldots,Z_{\ell(d)})\in\mathbb{Q}\langle\mathcal{A}\rangle\llbracket
Z_{1},\ldots,Z_{\ell(d)}\rrbracket.$

For some very well-behaved quasi-shuffle products appearing for example in the
theory of multiple zeta values and multiple q-zeta values, it is possible to
describe this product on the generating series of words by a recursive
explicit formula with respect to concatenation. This will explain the origin
of a particular kind of symmetries occurring in Ecalle’s theory of bimoulds
([Ec11]). Some important examples related to multiple q-zeta values will be
discussed in the following sections, for some other examples also related to
multiple zeta values, we refer to [Bu23, Appendix A.4 + A.5].
To translate the quasi-shuffle products into symmetries among bimoulds, we
apply a $\mathbb{Q}$-linear map
$\varphi:\mathbb{Q}\langle\mathcal{A}\rangle\to R$ into some
$\mathbb{Q}$-algebra $R$ to the first component of such a generating series of
words $\rho_{\mathcal{A}}(\mathcal{W})$. In other words, we consider the image
of the generic diagonal series $\mathcal{W}(\mathcal{A})$ under
$\varphi\otimes\rho_{\mathcal{A}}$.

######  Definition 3.5.

Let
$\rho_{\mathcal{A}}:\mathbb{Q}\langle\mathcal{A}\rangle\to\mathbb{Q}[Z_{1},Z_{2},\ldots]$
be a $\mathbb{Q}$-linear map as in Definition 3.1, $R$ be a
$\mathbb{Q}$-algebra and $\varphi:\mathbb{Q}\langle\mathcal{A}\rangle\to R$ be
a $\mathbb{Q}$-linear map. Then the _(commutative) generating series with
coefficients in $R$_ associated to $(\varphi,\rho_{\mathcal{A}})$ are given by

$(\varphi\otimes\rho_{\mathcal{A}})(\mathcal{W})_{d}(Z_{1},\ldots,Z_{\ell(d)})=\sum_{w\in(\mathcal{A}^{*})^{(d)}}\varphi(w)\rho_{\mathcal{A}}(w)\in
R\llbracket Z_{1},\ldots,Z_{\ell(d)}\rrbracket,\qquad d\geq 0.$

As before, we will often drop the depth index and simply write
$(\varphi\otimes\rho_{\mathcal{A}})(\mathcal{W})(Z_{1},\ldots,Z_{\ell(d)})$.
Such kind of sequences
$(\varphi\otimes\rho_{\mathcal{A}})(\mathcal{W})=\Big{(}(\varphi\otimes\rho_{\mathcal{A}})(\mathcal{W})_{d}\Big{)}_{d\geq
0}$ will occur as examples of Ecalle’s (bi-)moulds ([Ec11]).

######  Definition 3.6.

Let $R$ be a $\mathbb{Q}$-algebra. A sequence
$M=\left(M_{d}(X_{1},\ldots,X_{d})\right)_{d\geq
0}=\left(M_{0}(\emptyset),M_{1}(X_{1}),M_{2}(X_{1},X_{2}),\ldots\right)\in\prod_{d\geq
0}R\llbracket X_{1},\ldots,X_{d}\rrbracket$
is called a _mould_ with coefficients in $R$. Similarly, a sequence
$M=\left(M_{d}\binom{X_{1},\ldots,X_{d}}{Y_{1},\ldots,Y_{d}}\right)_{d\geq
0}=\left(M_{0}(\emptyset),M_{1}\binom{X_{1}}{Y_{1}},M_{2}\binom{X_{1},X_{2}}{Y_{1},Y_{2}},\ldots\right)\in\prod_{d\geq
0}R\llbracket X_{1},Y_{1},\ldots,X_{d},Y_{d}\rrbracket$
is called a _bimould_ with coefficients in $R$.

If
$\rho_{\mathcal{A}}:\mathbb{Q}\langle\mathcal{A}\rangle\to\mathbb{Q}[X_{1},X_{2},\ldots]$
is a map as in Definition 3.1 with $\ell(d)=d$ for all $d\geq 0$, then the
corresponding generating series of words
$\rho_{\mathcal{A}}(\mathcal{W})=(\rho_{\mathcal{A}}(\mathcal{W})_{d})_{d\geq
0}$ are a mould with coefficients in $\mathbb{Q}\langle\mathcal{A}\rangle$.
For any $\mathbb{Q}$-linear map
$\varphi:\mathbb{Q}\langle\mathcal{A}\rangle\to R$, the generating series
$(\varphi\otimes\rho_{\mathcal{A}})(\mathcal{W})=((\varphi\otimes\rho_{\mathcal{A}})(\mathcal{W})_{d})_{d\geq
0}$ associated to $(\varphi,\rho_{\mathcal{A}})$ are a mould with coefficients
in $R$. Similarly, if
$\rho_{\mathcal{A}}:\mathbb{Q}\langle\mathcal{A}\rangle\to\mathbb{Q}[X_{1},Y_{1},X_{2},Y_{2},\ldots]$
is a map as in Definition 3.1 with $\ell(d)=2d$ for every $d\geq 0$, then the
corresponding generating series of words
$\rho_{\mathcal{A}}(\mathcal{W})=(\rho_{\mathcal{A}}(\mathcal{W})_{d})_{d\geq
0}$ are a bimould with coefficients in the algebra
$\mathbb{Q}\langle\mathcal{A}\rangle$, and also the generating series
$(\varphi\otimes\rho_{\mathcal{A}})(\mathcal{W})=((\varphi\otimes\rho_{\mathcal{A}})(\mathcal{W})_{d})_{d\geq
0}$ associated to $(\varphi,\rho_{\mathcal{A}})$ for each $\mathbb{Q}$-linear
map $\varphi:\mathbb{Q}\langle\mathcal{A}\rangle\to R$ are a bimould with
coefficients in $R$.

######  Definition 3.7.

Let $R$ be a $\mathbb{Q}$-algebra and $\ell(0)<\ell(1)<\ell(2)<\dots$ a
sequence of non-negative integers. A sequence $M=(M_{d})_{d\geq 0}$ in
$\prod_{d\geq 0}R\llbracket Z_{1},\ldots,Z_{\ell(d)}\rrbracket$ is called
_$(\varphi_{\ast_{\diamond}},\rho_{\mathcal{A}})$ -symmetric_ if there exists
a $\mathbb{Q}$-algebra morphism
$\varphi_{\ast_{\diamond}}:(\mathbb{Q}\langle\mathcal{A}\rangle,\ast_{\diamond})\to
R$ and a $\mathbb{Q}$-linear map
$\rho_{\mathcal{A}}:\mathbb{Q}\langle\mathcal{A}\rangle\to\mathbb{Q}\llbracket
Z_{1},Z_{2},\ldots\rrbracket$ satisfying the conditions in Definition 3.1,
such that for all $d\geq 0$

$M_{d}=(\varphi_{\ast_{\diamond}}\otimes\rho_{\mathcal{A}})(\mathcal{W})_{d}.$

In the following sections, we will consider two particular quasi-shuffle
algebras, which will give rise to the notion of symmetril and b-symmetril
bimoulds following Definition 3.7.
Let $M=(M_{d})_{d\geq 0}\in\prod_{d\geq 0}R\llbracket
Z_{1},\ldots,Z_{d}\rrbracket$ be such a
$(\varphi_{\ast_{\diamond}},\rho_{\mathcal{A}})$-symmetric sequence. Then, one
obtains immediately from the definition that for $0<n<d$

$\displaystyle
M_{n}(Z_{1},\ldots,Z_{\ell(n)})M_{d-n}(Z_{\ell(n)+1},\ldots,Z_{\ell(d)})$
(3.1) $\displaystyle\hskip
142.26378pt=\varphi_{\ast_{\diamond}}\Big{(}\rho_{\mathcal{A}}(\mathcal{W})_{n}(Z_{1},\ldots,Z_{\ell(n)})\ast_{\diamond}\rho_{\mathcal{A}}(\mathcal{W})_{d-n}(Z_{\ell(n)+1},\ldots,Z_{\ell(d)})\Big{)}.$

The right-hand side is an element in
$\mathbb{Q}\langle\mathcal{A}\rangle\llbracket
Z_{1},\ldots,Z_{\ell(d)}\rrbracket$, so the map
$\varphi_{\ast_{\diamond}}:\mathbb{Q}\langle\mathcal{A}\rangle\to R$ needs to
be extended by $\mathbb{Q}\llbracket Z_{1},Z_{2},\ldots\rrbracket$-linearity.
As mentioned before, in some special cases the right-hand side can be
described in terms of an explicit recursive formula.
Alternatively, we can first apply the evaluation map
$\varphi:\mathbb{Q}\langle\mathcal{A}\rangle\to R$ for some
$\mathbb{Q}$-algebra $R$ to the generic diagonal series
$\mathcal{W}(\mathcal{A})$.

######  Definition 3.8.

Let $R$ be a $\mathbb{Q}$-algebra and
$\varphi:\mathbb{Q}\langle\mathcal{A}\rangle\to R$ be a $\mathbb{Q}$-linear
map. Define the _(non-commutative) generating series with coefficients in $R$_
associated to $\varphi$ by

$\varphi(\mathcal{W}(\mathcal{A}))=\sum_{w\in\mathcal{A}^{*}}\varphi(w)w\in
R\langle\langle\mathcal{A}\rangle\rangle.$

Here $R\langle\langle\mathcal{A}\rangle\rangle$ denotes non-commutative
algebra of power series over $R$ generated by $\mathcal{A}$.

The generating series $\varphi(\mathcal{W}(\mathcal{A}))$ can also be
decomposed into its homogeneous depth components
$\varphi(\mathcal{W}(\mathcal{A}))_{d}$ for $d\geq 0$ (similar to Definition
3.2).

######  Proposition 3.9.

Let $R$ be a $\mathbb{Q}$-algebra and
$\varphi:\mathbb{Q}\langle\mathcal{A}\rangle\to R$ be a $\mathbb{Q}$-linear
map. Assume that the algebra
$(\mathbb{Q}\langle\mathcal{A}\rangle,\ast_{\diamond})$ is graded with
$\operatorname{deg}(a)\geq 1$ for all $a\in\mathcal{A}$, and denote by
$\Delta_{\ast_{\diamond}}$ the dual completed coproduct to $\ast_{\diamond}$.
The map $\varphi$ is an algebra morphism for the quasi-shuffle product
$\ast_{\diamond}$ if and only if $\varphi(\mathcal{W}(\mathcal{A}))$ is
grouplike for the coproduct $\Delta_{\ast_{\diamond}}$.

###### Proof.

By duality, we have

$\Delta_{\ast_{\diamond}}\Big{(}\varphi(\mathcal{W}(\mathcal{A}))\Big{)}=\Delta_{\ast_{\diamond}}\left(\sum_{w\in\mathcal{A}^{*}}\varphi(w)w\right)=\sum_{u,v\in\mathcal{A}^{*}}\varphi(u\ast_{\diamond}v)u\otimes
v.$

So $\varphi$ is an algebra morphism for the quasi-shuffle product
$\ast_{\diamond}$ if and only if

$\displaystyle\Delta_{\ast_{\diamond}}\Big{(}\varphi(\mathcal{W}(\mathcal{A}))\Big{)}=\sum_{u,v\in\mathcal{A}^{*}}\varphi(u)\varphi(v)u\otimes
v=\varphi(\mathcal{W}(\mathcal{A}))\otimes\varphi(\mathcal{W}(\mathcal{A})).$

∎

In particular, applying the map $\varphi$ involves always a dualization
process. Summarizing this section, we obtain the following diagram

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##  4\. Generating series of words in
$\mathbb{Q}\langle\mathcal{Y}^{\operatorname{bi}}\rangle$

Consider the alphabet $\mathcal{Y}^{\operatorname{bi}}=\\{y_{k,m}\mid k\geq
1,\ m\geq 0\\}$ and let the _weight_ and the _depth_ of a word in
$\mathbb{Q}\langle\mathcal{Y}^{\operatorname{bi}}\rangle$ be given by

$\operatorname{wt}(y_{k_{1},m_{1}}\dots
y_{k_{d},m_{d}})=k_{1}+\dots+k_{d}+m_{1}+\dots+m_{d},\qquad\operatorname{dep}(y_{k_{1},m_{1}}\dots
y_{k_{d},m_{d}})=d,$

for all $k_{1},\ldots,k_{d}\geq 1,\ m_{1},\ldots,m_{d}\geq 0$. Define the
$\mathbb{Q}$-linear map

$\displaystyle\rho_{\mathcal{Y}^{\operatorname{bi}}}:\mathbb{Q}\langle\mathcal{Y}^{\operatorname{bi}}\rangle$
$\displaystyle\to\mathbb{Q}[X_{1},Y_{1},X_{2},Y_{2},\ldots],$ (4.1)
$\displaystyle y_{k_{1},m_{1}}\dots y_{k_{d},m_{d}}$ $\displaystyle\mapsto
X_{1}^{k_{1}-1}\frac{Y_{1}^{m_{1}}}{m_{1}!}\dots
X_{d}^{k_{d}-1}\frac{Y_{d}^{m_{d}}}{m_{d}!},$ (4.2)

then $\rho_{\mathcal{Y}^{\operatorname{bi}}}$ satisfies the conditions in
Definition 3.1 with $\ell(d)=2d$ for all $d\geq 0$. The generating series of
words in $\mathbb{Q}\langle\mathcal{Y}^{\operatorname{bi}}\rangle$ associated
to $\rho_{\mathcal{Y}^{\operatorname{bi}}}$ are given by
$\rho_{\mathcal{Y}^{\operatorname{bi}}}(\mathcal{W})_{0}=\mathbf{1}$ and for
$d\geq 1$ by

$\displaystyle\rho_{\mathcal{Y}^{\operatorname{bi}}}(\mathcal{W})_{d}\binom{X_{1},\ldots,X_{d}}{Y_{1},\ldots,Y_{d}}=\sum_{\begin{subarray}{c}k_{1},\ldots,k_{d}\geq
1\\\ m_{1},\ldots,m_{d}\geq 0\end{subarray}}y_{k_{1},m_{1}}\dots
y_{k_{d},m_{d}}X_{1}^{k_{1}-1}\frac{Y_{1}^{m_{1}}}{m_{1}!}\dots
X_{d}^{k_{d}-1}\frac{Y_{d}^{m_{d}}}{m_{d}!}.$ (4.3)

The stuffle product $\ast$ on
$\mathbb{Q}\langle\mathcal{Y}^{\operatorname{bi}}\rangle$ is one of the well-
behaved quasi-shuffle products, where we can give an explicit recursive
formula on the level of generating series of words.

######  Theorem 4.1.

For the stuffle product $\ast$ on
$\mathbb{Q}\langle\mathcal{Y}^{\operatorname{bi}}\rangle$ (Example 2.2.2.),
one obtains for all $0<n<d$ that
$\mathbf{1}\ast\rho_{\mathcal{Y}^{\operatorname{bi}}}(\mathcal{W})_{n}=\rho_{\mathcal{Y}^{\operatorname{bi}}}(\mathcal{W})_{n}\ast\mathbf{1}=\rho_{\mathcal{Y}^{\operatorname{bi}}}(\mathcal{W})_{n}$
and
$\displaystyle\rho_{\mathcal{Y}^{\operatorname{bi}}}(\mathcal{W})\binom{X_{1},\ldots,X_{n}}{Y_{1},\ldots,Y_{n}}\ast\rho_{\mathcal{Y}^{\operatorname{bi}}}(\mathcal{W})\binom{X_{n+1},\ldots,X_{d}}{Y_{n+1},\ldots,Y_{d}}$
$\displaystyle=\rho_{\mathcal{Y}^{\operatorname{bi}}}(\mathcal{W})\binom{X_{1}}{Y_{1}}\cdot\left(\rho_{\mathcal{Y}^{\operatorname{bi}}}(\mathcal{W})\binom{X_{2},\ldots,X_{n}}{Y_{2},\ldots,Y_{n}}\ast\rho_{\mathcal{Y}^{\operatorname{bi}}}(\mathcal{W})\binom{X_{n+1},\ldots,X_{d}}{Y_{n+1},\ldots,Y_{d}}\right)$
$\displaystyle+\rho_{\mathcal{Y}^{\operatorname{bi}}}(\mathcal{W})\binom{X_{n+1}}{Y_{n+1}}\cdot\left(\rho_{\mathcal{Y}^{\operatorname{bi}}}(\mathcal{W})\binom{X_{1},\ldots,X_{n}}{Y_{1},\ldots,Y_{n}}\ast\rho_{\mathcal{Y}^{\operatorname{bi}}}(\mathcal{W})\binom{X_{n+2},\ldots,X_{d}}{Y_{n+2},\ldots,Y_{d}}\right)$
$\displaystyle+\frac{\rho_{\mathcal{Y}^{\operatorname{bi}}}(\mathcal{W})\binom{X_{1}}{Y_{1}+Y_{n+1}}-\rho_{\mathcal{Y}^{\operatorname{bi}}}(\mathcal{W})\binom{X_{n+1}}{Y_{1}+Y_{n+1}}}{X_{1}-X_{n+1}}\cdot\left(\rho_{\mathcal{Y}^{\operatorname{bi}}}(\mathcal{W})\binom{X_{2},\ldots,X_{n}}{Y_{2},\ldots,Y_{n}}\ast\rho_{\mathcal{Y}^{\operatorname{bi}}}(\mathcal{W})\binom{X_{n+2},\ldots,X_{d}}{Y_{n+2},\ldots,Y_{d}}\right).$

###### Proof.

First, consider the case $d=2$ and compute directly

$\displaystyle\rho_{\mathcal{Y}^{\operatorname{bi}}}(\mathcal{W})\binom{X_{1}}{Y_{1}}\ast\rho_{\mathcal{Y}^{\operatorname{bi}}}(\mathcal{W})\binom{X_{2}}{Y_{2}}=\sum_{\begin{subarray}{c}k_{1},k_{2}\geq
1\\\ m_{1},m_{2}\geq 0\end{subarray}}(y_{k_{1},m_{1}}\ast
y_{k_{2},m_{2}})X_{1}^{k_{1}-1}\frac{Y_{1}^{m_{1}}}{m_{1}!}X_{2}^{k_{2}-1}\frac{Y_{2}^{m_{2}}}{m_{2}!}$
$\displaystyle=\sum_{\begin{subarray}{c}k_{1},k_{2}\geq 1\\\ m_{1},m_{2}\geq
0\end{subarray}}\Big{(}y_{k_{1},m_{1}}y_{k_{2},m_{2}}+y_{k_{2},m_{2}}y_{k_{1},m_{1}}+y_{k_{1}+k_{2},m_{1}+m_{2}}\Big{)}X_{1}^{k_{1}-1}\frac{Y_{1}^{m_{1}}}{m_{1}!}X_{2}^{k_{2}-1}\frac{Y_{2}^{m_{2}}}{m_{2}!}$
$\displaystyle=\rho_{\mathcal{Y}^{\operatorname{bi}}}(\mathcal{W})\binom{X_{1},X_{2}}{Y_{1},Y_{2}}+\rho_{\mathcal{Y}^{\operatorname{bi}}}(\mathcal{W})\binom{X_{2},X_{1}}{Y_{2},Y_{1}}+\frac{\rho_{\mathcal{Y}^{\operatorname{bi}}}(\mathcal{W})\binom{X_{1}}{Y_{1}+Y_{2}}-\rho_{\mathcal{Y}^{\operatorname{bi}}}(\mathcal{W})\binom{X_{2}}{Y_{1}+Y_{2}}}{X_{1}-X_{2}},$

where the last step follows from simple power series manipulations. Since the
definition of the stuffle product $\ast$ as well as the above formula for the
generating series $\rho_{\mathcal{Y}^{\operatorname{bi}}}(\mathcal{W})$ is
recursive, the arbitrary depth case follows similarly by induction. ∎

Following Definition 3.7, the quasi-shuffle algebra
$(\mathbb{Q}\langle\mathcal{Y}^{\operatorname{bi}}\rangle,\ast)$ gives rise to
a symmetry among bimoulds, which was studied extensively in [Ec11] and [Sc15].

######  Definition 4.2.

A bimould $M=(M_{d})_{d\geq 0}\in\prod_{d\geq 0}R\llbracket
X_{1},Y_{1},\ldots,X_{d},Y_{d}\rrbracket$ with coefficients in some
$\mathbb{Q}$-algebra $R$ is _symmetril_ if there is an algebra morphism
$\varphi_{\ast}:(\mathbb{Q}\langle\mathcal{Y}^{\operatorname{bi}}\rangle,\ast)\to
R$, such that $M$ is
$(\rho_{\mathcal{Y}^{\operatorname{bi}}},\varphi_{\ast})$-symmetric, i.e., we
have $M_{0}=1$ and for all $d\geq 1$

$M_{d}\binom{X_{1},\ldots,X_{d}}{Y_{1},\ldots,Y_{d}}=\sum_{\begin{subarray}{c}k_{1},\ldots,k_{d}\geq
1\\\ m_{1},\ldots,m_{d}\geq
0\end{subarray}}\varphi_{\ast}(y_{k_{1},m_{1}}\dots
y_{k_{d},m_{d}})X_{1}^{k_{1}-1}\frac{Y_{1}^{m_{1}}}{m_{1}!}\dots
X_{d}^{k_{d}-1}\frac{Y_{d}^{m_{d}}}{m_{d}!}.$

We refer to $\varphi_{\ast}$ as the _coefficient map_ of $M$.

######  Example 4.3.

Applying the recursive formula for $\ast$ on generating series of words given
in Theorem 4.1, one obtains that Definition 4.2 is equivalent to the
symmetrility defined in [Sc15, p. 17ff]. For example, symmetrility in depths
$2$ and $3$ means

$\displaystyle M\binom{X_{1}}{Y_{1}}\cdot M\binom{X_{2}}{Y_{2}}=$
$\displaystyle\
M\binom{X_{1},X_{2}}{Y_{1},Y_{2}}+M\binom{X_{2},X_{1}}{Y_{2},Y_{1}}+\frac{M\binom{X_{1}}{Y_{1}+Y_{2}}-M\binom{X_{2}}{Y_{1}+Y_{2}}}{X_{1}-X_{2}},$
$\displaystyle M\binom{X_{1}}{Y_{1}}\cdot M\binom{X_{2},X_{3}}{Y_{2},Y_{3}}=$
$\displaystyle\
M\binom{X_{1},X_{2},X_{3}}{Y_{1},Y_{2},Y_{3}}+M\binom{X_{2},X_{1},X_{3}}{Y_{2},Y_{1},Y_{3}}+M\binom{X_{2},X_{3},X_{1}}{Y_{2},Y_{3},Y_{1}}$
$\displaystyle+\frac{M\binom{X_{1},X_{3}}{Y_{1}+Y_{2},Y_{3}}-M\binom{X_{2},X_{3}}{Y_{1}+Y_{2},Y_{3}}}{X_{1}-X_{2}}+\frac{M\binom{X_{2},X_{1}}{Y_{2},Y_{1}+Y_{3}}-M\binom{X_{2},X_{3}}{Y_{2},Y_{1}+Y_{3}}}{X_{1}-X_{3}}.$

The deconcatenation coproduct on
$\mathbb{Q}\langle\mathcal{Y}^{\operatorname{bi}}\rangle$ (see (2.1)) is
compatible with the translation map $\rho_{\mathcal{Y}^{\operatorname{bi}}}$,
a straight-forward computation gives the following.

######  Proposition 4.4.

For each $d\geq 0$, we have

$\displaystyle\Delta_{\operatorname{dec}}\rho_{\mathcal{Y}^{\operatorname{bi}}}(\mathcal{W})\binom{X_{1},\ldots,X_{d}}{Y_{1},\ldots,Y_{d}}=\sum_{i=0}^{d}\rho_{\mathcal{Y}^{\operatorname{bi}}}(\mathcal{W})\binom{X_{1},\ldots,X_{i}}{Y_{1},\ldots,Y_{i}}\otimes\rho_{\mathcal{Y}^{\operatorname{bi}}}(\mathcal{W})\binom{X_{i+1},\ldots,Y_{d}}{X_{i+1},\ldots,Y_{d}}.$

##  5\. Generating series of words in
$\mathbb{Q}\langle\mathcal{B}\rangle^{0}$

Consider the alphabet $\mathcal{B}=\\{b_{0},b_{1},b_{2},\ldots\\}$ and define
the _weight_ and the _depth_ of a word in
$\mathbb{Q}\langle\mathcal{B}\rangle$ by

$\operatorname{wt}(b_{s_{1}}\dots b_{s_{l}})=s_{1}+\dots+s_{l}+\\#\\{i\mid
s_{i}=0\\},\qquad\operatorname{dep}(b_{s_{1}}\dots b_{s_{l}})=l-\\#\\{i\mid
s_{i}=0\\},$

for all $s_{1},\ldots,s_{l}\geq 0$. Let $\rho_{\mathcal{B}}$ be the
$\mathbb{Q}$-linear map given by

$\displaystyle\rho_{\mathcal{B}}:\mathbb{Q}\langle\mathcal{B}\rangle$
$\displaystyle\to\mathbb{Q}[Y_{0},X_{1},Y_{1},X_{2},Y_{2},\ldots],$ (5.1)
$\displaystyle b_{0}^{m_{0}}b_{k_{1}}b_{0}^{m_{1}}\dots
b_{k_{d}}b_{0}^{m_{d}}$ $\displaystyle\mapsto
Y_{0}^{m_{0}}X_{1}^{k_{1}-1}Y_{1}^{m_{1}}\dots
X_{d}^{k_{d}-1}Y_{d}^{m_{d}}\quad(k_{1},\ldots,k_{d}\geq
1,m_{1},\ldots,m_{d}\geq 0),$ (5.2)

then $\rho_{\mathcal{B}}$ satisfies the conditions in Definition 3.1 with
$\ell(d)=2d+1$ for all $d\geq 0$. The generating series of words associated to
$\rho_{\mathcal{B}}$ are given by for $d\geq 0$

$\rho_{\mathcal{B}}(\mathcal{W})_{d}\binom{\hskip
14.22636ptX_{1},\ldots,X_{d}}{Y_{0};Y_{1},\ldots,Y_{d}}=\sum_{\begin{subarray}{c}k_{1},\ldots,k_{d}\geq
1\\\ m_{0},\ldots,m_{d}\geq
0\end{subarray}}b_{0}^{m_{0}}b_{k_{1}}b_{0}^{m_{1}}\dots
b_{k_{d}}b_{0}^{m_{d}}Y_{0}^{m_{0}}X_{1}^{k_{1}-1}Y_{1}^{m_{1}}\dots
X_{d}^{k_{d}-1}Y_{d}^{m_{d}}.$

Let $\mathbb{Q}\langle\mathcal{B}\rangle^{0}$ be the subspace of
$\mathbb{Q}\langle\mathcal{B}\rangle$ generated by all words not starting in
$b_{0}$, i.e., we have

$\mathbb{Q}\langle\mathcal{B}\rangle^{0}=\mathbb{Q}\langle\mathcal{B}\rangle\backslash
b_{0}\mathbb{Q}\langle\mathcal{B}\rangle.$

Define the corresponding diagonal series

$\mathcal{W}(\mathcal{B})^{0}=\sum_{w\in\mathcal{B}^{*}\cap\mathbb{Q}\langle\mathcal{B}\rangle^{0}}w\otimes
w.$

Let
$\rho_{\mathcal{B}}^{0}:\mathbb{Q}\langle\mathcal{B}\rangle^{0}\to\mathbb{Q}[X_{1},Y_{1},X_{2},Y_{2},\ldots]$
be the restriction of $\rho_{\mathcal{B}}$ to
$\mathbb{Q}\langle\mathcal{B}\rangle^{0}$. Then the generating series of words
in $\mathbb{Q}\langle\mathcal{B}\rangle^{0}$ associated to
$\rho_{\mathcal{B}}^{0}$ are given by
$\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})_{0}=\mathbf{1}$ and

$\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})_{d}\binom{X_{1},\ldots,X_{d}}{Y_{1},\ldots,Y_{d}}=\sum_{\begin{subarray}{c}k_{1},\ldots,k_{d}\geq
1\\\ m_{1},\ldots,m_{d}\geq 0\end{subarray}}b_{k_{1}}b_{0}^{m_{1}}\dots
b_{k_{d}}b_{0}^{m_{d}}X_{1}^{k_{1}-1}Y_{1}^{m_{1}}\dots
X_{d}^{k_{d}-1}Y_{d}^{m_{d}},\quad d\geq 1.$

On $\mathbb{Q}\langle\mathcal{B}\rangle^{0}$ the balanced quasi-shuffle
product $\ast_{b}$ is very well-behaved, thus we can also give a recursive
definition on generating series of words for this product.

######  Theorem 5.1.

For the balanced quasi-shuffle product $\ast_{b}$ (Example 2.2.3.), one
obtains for all $0<n<d$ that
$\mathbf{1}\ast_{b}\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})_{n}=\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})_{n}\ast_{b}\
\mathbf{1}=\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})_{n}$ and
$\displaystyle\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{1},\ldots,X_{n}}{Y_{1},\ldots,Y_{n}}\ast_{b}\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{n+1},\ldots,X_{d}}{Y_{n+1},\ldots,Y_{d}}$
$\displaystyle=\left(\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{1},\ldots,X_{n-1}}{Y_{1},\ldots,Y_{n-1}}\ast_{b}\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{n+1},\ldots,X_{d}}{Y_{n+1},\ldots,Y_{d}}\right)\cdot\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{n}}{Y_{n}+Y_{d}}$
$\displaystyle+\left(\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{1},\ldots,X_{n}}{Y_{1},\ldots,Y_{n}}\ast_{b}\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{n+1},\ldots,X_{d-1}}{Y_{n+1},\ldots,Y_{d-1}}\right)\cdot\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{d}}{Y_{n}+Y_{d}}$
$\displaystyle+\left(\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{1},\ldots,X_{n-1}}{Y_{1},\ldots,Y_{n-1}}\ast_{b}\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{n+1},\ldots,X_{d-1}}{Y_{n+1},\ldots,Y_{d-1}}\right)\cdot\frac{\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{n}}{Y_{n}+Y_{d}}-\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{d}}{Y_{n}+Y_{d}}}{X_{n}-X_{d}}.$

###### Proof.

Consider the case $d=2$. We obtain
$\displaystyle\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{1}}{Y_{1}}$
$\displaystyle=\left(\sum_{k\geq
1}b_{k}X_{1}^{k-1}\right)\cdot\frac{\mathbf{1}}{\mathbf{1}-b_{0}Y_{1}}$ (5.3)
$\displaystyle=\sum_{k\geq
1}b_{k}X_{1}^{k-1}+\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{1}}{Y_{1}}\cdot
b_{0}Y_{1}.$ (5.4)
By applying the second identity in (5.3) and the definition of the balanced
quasi-shuffle product (we use the recursive definition from the right), we
obtain
$\displaystyle\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{1}}{Y_{1}}\ast_{b}\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{2}}{Y_{2}}$
$\displaystyle=\left(\sum_{k\geq
1}b_{k}X_{1}^{k-1}\right)\ast_{b}\left(\sum_{k\geq
1}b_{k}X_{2}^{k-1}\right)+\left(\sum_{k\geq
1}b_{k}X_{1}^{k-1}\right)\ast_{b}\left(\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{2}}{Y_{2}}\cdot
b_{0}Y_{2}\right)\hskip 85.35826pt$
$\displaystyle+\left(\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{1}}{Y_{1}}\cdot
b_{0}Y_{1}\right)\ast_{b}\left(\sum_{k\geq
1}b_{k}X_{2}^{k-1}\right)+\left(\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{1}}{Y_{1}}\cdot
b_{0}Y_{1}\right)\ast_{b}\left(\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{2}}{Y_{2}}\cdot
b_{0}Y_{2}\right)$
$\displaystyle=\left(\sum_{k\geq
1}b_{k}X_{1}^{k-1}\right)\ast_{b}\left(\sum_{k\geq
1}b_{k}X_{2}^{k-1}\right)+\left(\left(\sum_{k\geq
1}b_{k}X_{1}^{k-1}\right)\ast_{b}\left(\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{2}}{Y_{2}}Y_{2}\right)\right)\cdot
b_{0}$
$\displaystyle+\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{2}}{Y_{2}}\cdot
b_{0}Y_{2}\cdot\sum_{k\geq
1}b_{k}X_{1}^{k-1}+\left(\left(\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{1}}{Y_{1}}Y_{1}\right)\ast_{b}\left(\sum_{k\geq
1}b_{k}X_{2}^{k-1}\right)\right)\cdot b_{0}$
$\displaystyle+\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{1}}{Y_{1}}\cdot
b_{0}Y_{1}\cdot\sum_{k\geq
1}b_{k}X_{2}^{k-1}+\left(\left(\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{1}}{Y_{1}}Y_{1}\right)\ast_{b}\left(\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{2}}{Y_{2}}\cdot
b_{0}Y_{2}\right)\right)\cdot b_{0}$
$\displaystyle+\left(\left(\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{1}}{Y_{1}}\cdot
b_{0}Y_{1}\right)\ast_{b}\left(\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{2}}{Y_{2}}Y_{2}\right)\right)\cdot
b_{0}.$
Applying again the second identity in (5.3) together with some cancellation,
we deduce
$\displaystyle\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{1}}{Y_{1}}\ast_{b}\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{2}}{Y_{2}}$
$\displaystyle=\left(\sum_{k\geq
1}b_{k}X_{1}^{k-1}\right)\ast_{b}\left(\sum_{k\geq
1}b_{k}X_{2}^{k-1}\right)-2\left(\sum_{k\geq
1}b_{k}X_{1}^{k-1}\right)\cdot\left(\sum_{k\geq
1}b_{k}X_{2}^{k-1}\right)+\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{2}}{Y_{2}}\cdot\sum_{k\geq
1}b_{k}X_{1}^{k-1}$
$\displaystyle+\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{1}}{Y_{1}}\cdot\sum_{k\geq
1}b_{k}X_{2}^{k-1}+\left(\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{1}}{Y_{1}}\ast_{b}\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{2}}{Y_{2}}\right)\cdot
b_{0}(Y_{1}+Y_{2}).$
Using the first identity in (5.3) together with the formula for the
concatenation product in terms of generating series of words (Proposition
3.4), we obtain
$\displaystyle\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{1}}{Y_{1}}\ast_{b}\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{2}}{Y_{2}}$
(5.5) $\displaystyle=\left(\left(\sum_{k\geq
1}b_{k}X_{1}^{k-1}\right)\ast_{b}\left(\sum_{k\geq
1}b_{k}X_{2}^{k-1}\right)-2\left(\sum_{k\geq
1}b_{k}X_{1}^{k-1}\right)\cdot\left(\sum_{k\geq
1}b_{k}X_{2}^{k-1}\right)\right)\cdot\frac{\mathbf{1}}{\mathbf{1}-b_{0}(Y_{1}+Y_{2})}$
$\displaystyle\hskip
12.80365pt+\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{2},X_{1}}{Y_{2},Y_{1}+Y_{2}}+\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{1},X_{2}}{Y_{1},Y_{1}+Y_{2}}.$
Moreover, one verifies by applying the definition of $\ast_{b}$ and some power
series manipulation
$\displaystyle\left(\sum_{k\geq
1}b_{k}X_{1}^{k-1}\right)\ast_{b}\left(\sum_{k\geq
1}b_{k}X_{2}^{k-1}\right)=2\left(\sum_{k\geq
1}b_{k}X_{1}^{k-1}\right)\cdot\left(\sum_{k\geq
1}b_{k}X_{2}^{k-1}\right)+\frac{\sum_{k\geq 1}b_{k}X_{1}^{k_{1}}-\sum_{k\geq
1}b_{k}X_{2}^{k-1}}{X_{1}-X_{2}}.$
Applying this result to (5.5) together with (5.3) gives
$\displaystyle\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{1}}{Y_{1}}\ast_{b}\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{2}}{Y_{2}}$
$\displaystyle\hskip
51.21504pt=\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{2},X_{1}}{Y_{2},Y_{1}+Y_{2}}+\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{1},X_{2}}{Y_{1},Y_{1}+Y_{2}}+\frac{\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{1}}{Y_{1}+Y_{2}}-\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{2}}{Y_{1}+Y_{2}}}{X_{1}-X_{2}}.$
This equals exactly the claimed formula of $\ast_{b}$ for the generating
series in depth $2$. Since the definition of the product $\ast_{b}$ and the
above generating series formula are both recursive, we obtain the desired
formula in arbitrary depth by applying induction and the same computations as
before. ∎

The quasi-shuffle algebra $(\mathbb{Q}\langle\mathcal{B}\rangle^{0},\ast_{b})$
gives rise to symmetry among bimoulds according to Definition 3.7.

######  Definition 5.2.

A bimould $M=(M_{d})_{d\geq 0}\in\prod_{d\geq 0}R\llbracket
X_{1},Y_{1},\ldots,X_{d},Y_{d}\rrbracket$ with coefficients in some
$\mathbb{Q}$-algebra $R$ is _b-symmetril_ if there is an algebra morphism
$\varphi_{\ast_{b}}:(\mathbb{Q}\langle\mathcal{B}\rangle^{0},\ast_{b})\to R$,
such that $M$ is $(\varphi_{\ast_{b}},\rho_{\mathcal{B}}^{0})$-symmetric,
i.e., we have $M_{0}=1$ and for all $d\geq 1$

$M_{d}\binom{X_{1},\ldots,X_{d}}{Y_{1},\ldots,Y_{d}}=\sum_{\begin{subarray}{c}k_{1},\ldots,k_{d}\geq
1\\\ m_{1},\ldots,m_{d}\geq
0\end{subarray}}\varphi_{\ast_{b}}(b_{k_{1}}b_{0}^{m_{1}}\dots
b_{k_{d}}b_{0}^{m_{d}})X_{1}^{k_{1}-1}Y_{1}^{m_{1}}\dots
X_{d}^{k_{d}-1}Y_{d}^{m_{d}}.$

As before, we call the map $\varphi_{\ast_{b}}$ the _coefficient map_ of $M$.

######  Example 5.3.

Applying the recursive formula for $\ast_{b}$ on the generating series of
words given in Theorem 5.1, we obtain that b-symmetrility in depth $2$ and $3$
means

$\displaystyle M\binom{X_{1}}{Y_{1}}\cdot M\binom{X_{2}}{Y_{2}}=$
$\displaystyle\
M\binom{X_{2},X_{1}}{Y_{2},Y_{1}+Y_{2}}+M\binom{X_{1},X_{2}}{Y_{1},Y_{1}+Y_{2}}+\frac{M\binom{X_{1}}{Y_{1}+Y_{2}}-M\binom{X_{2}}{Y_{1}+Y_{2}}}{X_{1}-X_{2}},$
$\displaystyle M\binom{X_{1}}{Y_{1}}\cdot M\binom{X_{2},X_{3}}{Y_{2},Y_{3}}=$
$\displaystyle\
M\binom{X_{2},X_{3},X_{1}}{Y_{2},Y_{3},Y_{1}+Y_{3}}+M\binom{X_{2},X_{1},X_{3}}{Y_{2},Y_{1}+Y_{2},Y_{1}+Y_{3}}$
$\displaystyle+M\binom{X_{1},X_{2},X_{3}}{Y_{1},Y_{1}+Y_{2},Y_{1}+Y_{3}}+\frac{M\binom{X_{2},X_{1}}{Y_{2},Y_{1}+Y_{3}}-M\binom{X_{2},X_{3}}{Y_{2},Y_{1}+Y_{3}}}{X_{1}-X_{3}}$
$\displaystyle+\frac{M\binom{X_{1},X_{3}}{Y_{1}+Y_{2},Y_{1}+Y_{3}}-M\binom{X_{2},X_{3}}{Y_{1}+Y_{2},Y_{1}+Y_{3}}}{X_{1}-X_{2}}.$

##  6\. Generating series of words in $\mathbb{Q}\langle\mathcal{B}\rangle$
and regularization

The deconcatenation coproduct on $\mathbb{Q}\langle\mathcal{B}\rangle$
(defined in (2.1)) is compatible with the translation map
$\rho_{\mathcal{B}}$, hence it can be described in terms of the generating
series $\rho_{\mathcal{B}}(\mathcal{W})$ of words in
$\mathbb{Q}\langle\mathcal{B}\rangle$. But the deconcatenation coproduct does
not preserve the subspace $\mathbb{Q}\langle\mathcal{B}\rangle^{0}$.
Therefore, we will introduce a regularization map and a regularized coproduct
and describe both in terms of the generating series
$\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})$ of words in
$\mathbb{Q}\langle\mathcal{B}\rangle^{0}$.

######  Proposition 6.1.

For all $d\geq 0$, we have

$\Delta_{\operatorname{dec}}\rho_{\mathcal{B}}(\mathcal{W})\binom{\hskip
14.22636ptX_{1},\ldots,X_{d}}{Y_{0};Y_{1},\ldots,Y_{d}}=\sum_{i=0}^{d}\rho_{\mathcal{B}}(\mathcal{W})\binom{\hskip
14.22636ptX_{1},\ldots,X_{i}}{Y_{0};Y_{1},\ldots,Y_{i}}\otimes\rho_{\mathcal{B}}(\mathcal{W})\binom{\hskip
14.22636ptX_{i+1},\ldots,X_{d}}{Y_{i};Y_{i+1},\ldots,Y_{d}}.$

###### Proof.

By definition of the deconcatenation coproduct, we have
$\displaystyle\Delta_{\operatorname{dec}}\rho_{\mathcal{B}}(\mathcal{W})\binom{\hskip
14.22636ptX_{1},\ldots,X_{d}}{Y_{0};Y_{1},\ldots,Y_{d}}$
$\displaystyle=\sum_{\begin{subarray}{c}k_{1},\dots,k_{d}\geq 1\\\
m_{0},\dots,m_{d}\geq
0\end{subarray}}\Delta_{\operatorname{dec}}(b_{0}^{m_{0}}b_{k_{1}}b_{0}^{m_{1}}\dots
b_{k_{d}}b_{0}^{m_{d}})Y_{0}^{m_{0}}X_{1}^{k_{1}-1}Y_{1}^{m_{1}}\dots
X_{d}^{k_{d}-1}Y_{d}^{m_{d}}$
$\displaystyle=\sum_{i=0}^{d}\sum_{\begin{subarray}{c}k_{1},\dots,k_{d}\geq
1\\\ m_{0},\dots,m_{d}\geq
0\end{subarray}}\sum_{\begin{subarray}{c}n_{1}+n_{2}=m_{i}\\\ n_{1},n_{2}\geq
0\end{subarray}}(b_{0}^{m_{0}}b_{k_{1}}b_{0}^{m_{1}}\dots
b_{k_{i}}b_{0}^{n_{1}}\otimes b_{0}^{n_{2}}b_{k_{i+1}}b_{0}^{m_{i+1}}\dots
b_{k_{d}}b_{0}^{m_{d}})Y_{0}^{m_{0}}X_{1}^{k_{1}-1}Y_{1}^{m_{1}}\dots
X_{d}^{k_{d}-1}Y_{d}^{m_{d}}$ Reordering these sums, we obtain that this
expression is equal to
$\displaystyle\sum_{i=0}^{d}\sum_{\begin{subarray}{c}k_{1},\dots,k_{d}\geq
1\\\ m_{0},\dots,m_{i-1},n_{1},n_{2},m_{i+1},\ldots,m_{d}\geq
0\end{subarray}}(b_{0}^{m_{0}}b_{k_{1}}b_{0}^{m_{1}}\dots
b_{k_{i}}b_{0}^{n_{1}}\otimes b_{0}^{n_{2}}b_{k_{i+1}}b_{0}^{m_{i+1}}\dots
b_{k_{d}}b_{0}^{m_{d}})Y_{0}^{m_{0}}X_{1}^{k_{1}-1}Y_{1}^{m_{1}}$
$\displaystyle\hskip 270.30118pt\dots
X_{i}^{k_{i}-1}Y_{i}^{n_{1}+n_{2}}X_{i+1}^{k_{i+1}-1}Y_{i+1}^{m_{i+1}}\dots
X_{d}^{k_{d}-1}Y_{d}^{m_{d}}$
$\displaystyle=\sum_{i=0}^{d}\rho_{\mathcal{B}}(\mathcal{W})\binom{\hskip
14.22636ptX_{1},\ldots,X_{i}}{Y_{0};Y_{1},\ldots,Y_{i}}\otimes\rho_{\mathcal{B}}(\mathcal{W})\binom{\hskip
14.22636ptX_{i+1},\ldots,X_{d}}{Y_{i};Y_{i+1},\ldots,Y_{d}}$
∎

Next, we introduce a regularization, which assigns to each word in
$\mathbb{Q}\langle\mathcal{B}\rangle$ an element in
$\mathbb{Q}\langle\mathcal{B}\rangle^{0}$, such that the balanced quasi-
shuffle product $\ast_{b}$ is preserved. The regularization process is similar
to the ones for multiple zeta values given in [IKZ06, Proposition 1]. We will
rephrase this regularization map in terms of generating series of words.

######  Proposition 6.2.

Let $T$ be a formal variable and extend the product $\ast_{b}$ by
$\mathbb{Q}[T]$-linearity to $\mathbb{Q}\langle\mathcal{B}\rangle^{0}[T]$. The
map

$\displaystyle\operatorname{reg}_{T}:\
\mathbb{Q}\langle\mathcal{B}\rangle^{0}[T]$
$\displaystyle\to\mathbb{Q}\langle\mathcal{B}\rangle,$ $\displaystyle wT^{n}$
$\displaystyle\mapsto w\ast_{b}b_{0}^{\ast_{b}n}$

is an algebra isomorphism for the balanced quasi-shuffle product $\ast_{b}$.

###### Proof.

For the surjectivity of $\operatorname{reg}_{T}$, we show that any word
$w\in\mathbb{Q}\langle\mathcal{B}\rangle$ is a polynomial in $b_{0}$ with
coefficients in $\mathbb{Q}\langle\mathcal{B}\rangle^{0}$. Let
$w=b_{0}^{m_{0}}b_{k_{1}}b_{0}^{m_{1}}\dots b_{k_{d}}b_{0}^{m_{d}}$ for some
integers $k_{1},\ldots,k_{d}\geq 1$ and $m_{0},\ldots,m_{d}\geq 0$. We prove
by induction on $m_{0}$, that $w=u+v\ast_{b}b_{0}$ for some
$u\in\mathbb{Q}\langle\mathcal{B}\rangle^{0}$ and
$v\in\mathbb{Q}\langle\mathcal{B}\rangle$, where all words in $v$ have weight
smaller than $\operatorname{wt}(w)$. Then induction on the weight proves the
claim. The case $m_{0}=0$ is trivial, simply choose $u=w,\ v=0$. Next,
calculate

$\displaystyle b_{0}^{m_{0}-1}b_{k_{1}}b_{0}^{m_{1}}\dots
b_{k_{d}}b_{0}^{m_{d}}\ast_{b}b_{0}=$ $\displaystyle\ m_{0}\
b_{0}^{m_{0}}b_{k_{1}}b_{0}^{m_{1}}\dots b_{k_{d}}b_{0}^{m_{d}}$
$\displaystyle+\sum_{i=1}^{d}(m_{i}+1)\
b_{0}^{m_{0}-1}b_{k_{1}}b_{0}^{m_{1}}\dots
b_{k_{i}}b_{0}^{m_{i}+1}b_{k_{i+1}}b_{0}^{m_{i+1}}\dots
b_{k_{d}}b_{0}^{m_{d}}.$

Applying the induction hypotheses to every word in the second line leads to

$w=\frac{1}{m_{0}}\left(u+\big{(}v+b_{0}^{m_{0}-1}b_{k_{1}}b_{0}^{m_{1}}\dots
b_{k_{d}}b_{0}^{m_{d}}\big{)}\ast_{b}b_{0}\right)$

for some $u\in\mathbb{Q}\langle\mathcal{B}\rangle^{0}$ and
$v\in\mathbb{Q}\langle\mathcal{B}\rangle$, where
$v+b_{0}^{m_{0}-1}b_{k_{1}}b_{0}^{m_{1}}\dots b_{k_{d}}b_{0}^{m_{d}}$ consists
of words of weight smaller than $\operatorname{wt}(w)$.
Let $P\in\mathbb{Q}\langle\mathcal{B}\rangle^{0}[T]\backslash\\{0\\}$ and
write $P=wT^{n}+R$, where
$w\in\mathbb{Q}\langle\mathcal{B}\rangle^{0}\backslash\\{0\\}$ and
$R\in\mathbb{Q}\langle\mathcal{B}\rangle^{0}[T]$ is a polynomial of degree
smaller than $n$. We have

$\operatorname{reg}_{T}(P)=w\ast_{b}b_{0}^{\ast_{b}n}+\operatorname{reg}_{T}(R)=n!b_{0}^{n}w+\widetilde{w},$

where $\widetilde{w}\in\mathbb{Q}\langle\mathcal{B}\rangle$ consists of words
with at most $(n-1)$-times the letter $b_{0}$ at the beginning. We deduce
$\operatorname{reg}_{T}(P)\neq 0$, thus $\operatorname{reg}_{T}$ is injective.
∎

In particular, by first applying the inverse
$\operatorname{reg}_{T}^{-1}:\mathbb{Q}\langle\mathcal{B}\rangle\to\mathbb{Q}\langle\mathcal{B}\rangle^{0}[T]$
and then evaluating in $T=0$ yields the desired _regularization map_

$\displaystyle\operatorname{reg}:(\mathbb{Q}\langle\mathcal{B}\rangle,\ast_{b})\to(\mathbb{Q}\langle\mathcal{B}\rangle^{0},\ast_{b}).$
(6.1)

By construction, it is an algebra morphism for the balanced quasi-shuffle
product $\ast_{b}$.

######  Theorem 6.3.

For all $d\geq 1$, we have

$\displaystyle\operatorname{reg}\rho_{\mathcal{B}}(\mathcal{W})\binom{\hskip
14.22636ptX_{1},\ldots,X_{d}}{Y_{0};Y_{1},\ldots,Y_{d}}=\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{1},\ldots,X_{d}}{Y_{1}-Y_{0},\ldots,Y_{d}-Y_{0}}.$

###### Proof.

We obtain from [IKZ06, equation (5.2)] applied to our setup that for each word
$w=b_{0}^{m_{0}}b_{k_{1}}b_{0}^{m_{1}}\dots b_{k_{d}}b_{0}^{m_{d}}$ we have

$\displaystyle\operatorname{reg}(w)$
$\displaystyle=(-1)^{m_{0}}b_{k_{1}}(b_{0}^{m_{0}}\ast_{b}b_{0}^{m_{1}}b_{k_{2}}b_{0}^{m_{2}}\dots
b_{k_{d}}b_{0}^{m_{d}})$
$\displaystyle=(-1)^{m_{0}}\sum_{\begin{subarray}{c}n_{1}+\dots+n_{d}=m_{0}\\\
n_{1},\ldots,n_{d}\geq
0\end{subarray}}\binom{m_{1}+n_{1}}{n_{1}}\dots\binom{m_{d}+n_{d}}{n_{d}}b_{k_{1}}b_{0}^{m_{1}+n_{1}}b_{k_{2}}b_{0}^{m_{2}+n_{2}}\dots
b_{k_{d}}b_{0}^{m_{d}+n_{d}}.$

Hence we deduce

$\displaystyle\operatorname{reg}\rho_{\mathcal{B}}(\mathcal{W})\binom{\hskip
14.22636ptX_{1},\ldots,X_{d}}{Y_{0};Y_{1},\ldots,Y_{d}}$ (6.2)
$\displaystyle=\sum_{\begin{subarray}{c}k_{1},\dots,k_{d}\geq 1\\\
m_{0},\dots,m_{d}\geq
0\end{subarray}}(-1)^{m_{0}}\sum_{\begin{subarray}{c}n_{1}+\dots+n_{d}=m_{0}\\\
n_{1},\ldots,n_{d}\geq
0\end{subarray}}\binom{m_{1}+n_{1}}{n_{1}}\dots\binom{m_{d}+n_{d}}{n_{d}}b_{k_{1}}b_{0}^{m_{1}+n_{1}}b_{k_{2}}b_{0}^{m_{2}+n_{2}}\dots
b_{k_{d}}b_{0}^{m_{d}+n_{d}}$ $\displaystyle\hskip 312.9803pt\cdot
Y_{0}^{m_{0}}X_{1}^{k_{1}-1}Y_{1}^{m_{1}}\dots X_{d}^{k_{d}-1}Y_{d}^{m_{d}}.$

On the other hand, we compute

$\displaystyle\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{1},\ldots,X_{d}}{Y_{1}-Y_{0},\ldots,Y_{d}-Y_{0}}$
(6.4) $\displaystyle=\sum_{\begin{subarray}{c}k_{1},\dots,k_{d}\geq 1\\\
m_{1},\dots,m_{d}\geq
0\end{subarray}}\sum_{n_{1}=0}^{m_{1}}\dots\sum_{n_{d}=0}^{m_{d}}(-1)^{n_{1}+\dots+n_{d}}\binom{m_{1}}{n_{1}}\dots\binom{m_{d}}{n_{d}}b_{k_{1}}b_{0}^{m_{1}}\dots
b_{k_{d}}b_{0}^{m_{d}}Y_{0}^{n_{1}+\dots+n_{d}}X_{1}^{k_{1}-1}Y_{1}^{m_{1}-n_{1}}$
$\displaystyle\hskip 378.42174pt\dots X_{d}^{k_{d}-1}Y_{d}^{m_{d}-n_{d}}.$

Comparing the coefficients of the monomial
$Y_{0}^{m_{0}}X_{1}^{k_{1}-1}Y_{1}^{m_{1}}\dots X_{d}^{k_{d}-1}Y_{d}^{m_{d}}$
in (6.2) and (6.4) yields the claimed equality. ∎

######  Definition 6.4.

On the space $\mathbb{Q}\langle\mathcal{B}\rangle^{0}$ define the _regularized
coproduct_

$\Delta_{\operatorname{dec}}^{0}=(\operatorname{reg}\otimes\operatorname{reg})\circ\Delta_{\operatorname{dec}}.$

Combining the formulas given in Proposition 6.1 and Theorem 6.3, we also
obtain a description of the regularized coproduct
$\Delta_{\operatorname{dec}}^{0}$ on the generating series
$\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})$ of words in
$\mathbb{Q}\langle\mathcal{B}\rangle^{0}$.

######  Proposition 6.5.

For all $d\geq 1$, we have

$\displaystyle\Delta_{\operatorname{dec}}^{0}\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{1},\ldots,X_{d}}{Y_{1},\ldots,Y_{d}}$
$\displaystyle=\sum_{i=0}^{d}\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{1},\ldots,X_{i}}{Y_{1},\ldots,Y_{i}}\otimes\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{i+1},\ldots,X_{d}}{Y_{i+1}-Y_{i},\ldots,Y_{d}-Y_{i}}.$

##  7\. Comparison of the generating series of words in
$\mathbb{Q}\langle\mathcal{Y}^{\operatorname{bi}}\rangle$ and
$\mathbb{Q}\langle\mathcal{B}\rangle^{0}$

In this section, we will relate the product and coproduct formulas given for
the generating series of words in
$\mathbb{Q}\langle\mathcal{Y}^{\operatorname{bi}}\rangle$ and
$\mathbb{Q}\langle\mathcal{B}\rangle^{0}$ (Section 4, 5, 6).

######  Definition 7.1.

For any bimould $M=(M_{d})_{d\geq 0}$ define the bimould
$M^{\\#_{Y}}=(M_{d}^{\\#_{Y}})_{d\geq 0}$ by

$M_{d}^{\\#_{Y}}\binom{X_{1},\ldots,X_{d}}{Y_{1},\ldots,Y_{d}}=M_{d}\binom{X_{1},\ldots,X_{d}}{Y_{1},Y_{1}+Y_{2},\ldots,Y_{1}+\dots+Y_{d}},\qquad
d\geq 0.$

Evidently, the inverse operation is given by

$M_{d}^{\\#_{Y}^{-1}}\binom{X_{1},\ldots,X_{d}}{Y_{1},\ldots,Y_{d}}=M_{d}\binom{X_{1},\ldots,X_{d}}{Y_{1},Y_{2}-Y_{1},\ldots,Y_{d}-Y_{d-1}},\qquad
d\geq 0.$

######  Definition 7.2.

Let
$\varphi_{\\#}:\mathbb{Q}\langle\mathcal{Y}^{\operatorname{bi}}\rangle\to\mathbb{Q}\langle\mathcal{B}\rangle^{0}$
be the $\mathbb{Q}$-linear map implicitly defined by

$\displaystyle(\varphi_{\\#}\otimes\rho_{\mathcal{Y}^{\operatorname{bi}}})(\mathcal{W})_{d}\binom{X_{1},\ldots,X_{d}}{Y_{1},\ldots,Y_{d}}=\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})_{d}^{\\#_{Y}}\binom{X_{1},\ldots,X_{d}}{Y_{1},\ldots,Y_{d}},$

i.e., the coefficient of $X_{1}^{k_{1}-1}\frac{Y_{1}^{m_{1}}}{m_{1}!}\dots
X_{d}^{k_{d}-1}\frac{Y_{d}^{m_{d}}}{m_{d}!}$ in
$\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})_{d}^{\\#_{Y}}$ equals the image of
the word $y_{k_{1},m_{1}}\dots y_{k_{d},m_{d}}$ under the map $\varphi_{\\#}$.

######  Example 7.3.

We obtain for all $k_{1},k_{2}\geq 1,\ m_{1},m_{2}\geq 0$

$\displaystyle\varphi_{\\#}(y_{k_{1},m_{1}})$
$\displaystyle=m_{1}!b_{k_{1}}b_{0}^{m_{1}},$
$\displaystyle\varphi_{\\#}(y_{k_{1},m_{1}}y_{k_{2},m_{2}})$
$\displaystyle=\sum_{n=m_{2}}^{m_{1}+m_{2}}\frac{m_{1}!n!}{(n-m_{2})!}b_{k_{1}}b_{0}^{m_{1}+m_{2}-n}b_{k_{2}}b_{0}^{n}.$

######  Theorem 7.4.

The map
$\varphi_{\\#}:(\mathbb{Q}\langle\mathcal{Y}^{\operatorname{bi}}\rangle,\ast)\to(\mathbb{Q}\langle\mathcal{B}\rangle^{0},\ast_{b})$
is an isomorphism of weight-graded algebras.

###### Proof.

Evidently, $\varphi_{\\#}$ is an isomorphism of graded vector
$\mathbb{Q}$-vector spaces. We will now show the compatibility of the products
$\ast$ and $\ast_{b}$ on the level of generating series, this means we will
prove that $\rho_{\mathcal{B}}(\mathcal{W})^{\\#_{Y}}$ satisfies the stuffle
product formula given in Theorem 4.1. For all $d,d^{\prime}\geq 1$ denote by
$\shuffle(d,d^{\prime})$ the set of all permutations $\sigma\in
S_{d+d^{\prime}}$ satisfying $\sigma(1)<\dots<\sigma(d)$,
$\sigma(d+1)<\dots<\sigma(d+d^{\prime})$. Moreover, write
$\underline{Y_{i}}=Y_{1}+\dots+Y_{i}$ for $1\leq i\leq d$ and
$\underline{Y_{d+j}}=Y_{d+1}+\dots+Y_{d+j}$ for $1\leq j\leq d^{\prime}$. Then
by Theorem 5.1, we have modulo terms of lower depths

$\displaystyle\rho_{\mathcal{B}}(\mathcal{W})^{\\#_{Y}}\binom{X_{1},\ldots,X_{d}}{Y_{1},\ldots,Y_{d}}\rho_{\mathcal{B}}(\mathcal{W})^{\\#_{Y}}\binom{X_{d+1},\ldots,X_{d+d^{\prime}}}{Y_{d+1},\ldots,Y_{d+d^{\prime}}}$
(7.1) $\displaystyle\hskip
113.81102pt\equiv\sum_{\sigma\in\shuffle(d,d^{\prime})}\rho_{\mathcal{B}}(\mathcal{W})\binom{X_{\sigma^{-1}(1)},\ldots,X_{\sigma^{-1}(d+d^{\prime})}}{\underline{Y_{\sigma^{-1}(1)}}+\underline{Y_{\sigma_{\mu}^{-1}(1)}},\ldots,\underline{Y_{\sigma^{-1}(d+d^{\prime})}}+\underline{Y_{\sigma_{\mu}^{-1}(d+d‘)}}}.$

Here we set

$\sigma_{\mu}^{-1}(k)=\begin{cases}\sigma^{-1}\big{(}\max\\{n\mid\sigma^{-1}(n)>d,n<k\\}\big{)},&1\leq\sigma^{-1}(k)\leq
d,\\\ \sigma^{-1}\big{(}\max\\{n\mid\sigma^{-1}(n)\leq d,\
n<k\\}\big{)},&d+1\leq\sigma^{-1}(k)\leq d+d^{\prime},\end{cases}$

and $\underline{Y_{\sigma_{\mu}^{-1}(k)}}=0$ if such a number $n$ does not
exists. Observe that for every term in the right-hand side of (7.1) and for
every $k=1,\ldots,d+d^{\prime}$ the predecessor of the entry
$\underline{Y_{\sigma^{-1}(k)}}+\underline{Y_{\sigma_{\mu}^{-1}(k)}}$ in the
bi-index is given by
$\underline{Y_{\sigma^{-1}(k)-1}}+\underline{Y_{\sigma_{\mu}^{-1}(k)}}$ (where
$\underline{Y_{\sigma^{-1}(k)-1}}:=0$ if $k\in\\{1,d+1\\}$) and, we have
$\underline{Y_{n}}-\underline{Y_{n-1}}=Y_{n}$. Thus, we deduce

$\displaystyle\rho_{\mathcal{B}}(\mathcal{W})^{\\#_{Y}^{-1}}\binom{X_{\sigma^{-1}(1)},\ldots,X_{\sigma^{-1}(d+d^{\prime})}}{\underline{Y_{\sigma^{-1}(1)}}+\underline{Y_{\sigma_{\mu}^{-1}(1)}},\ldots,\underline{Y_{\sigma^{-1}(d+d^{\prime})}}+\underline{Y_{\sigma_{\mu}^{-1}(d+d‘)}}}=\rho_{\mathcal{B}}(\mathcal{W})\binom{X_{\sigma^{-1}(1)},\ldots,X_{\sigma^{-1}(d+d^{\prime})}}{Y_{\sigma^{-1}(1)},\ldots,Y_{\sigma^{-1}(d+d^{\prime})}}.$
(7.2)

Combining the equations (7.1) and (7.2), we get modulo terms of lower depths

$\displaystyle\rho_{\mathcal{B}}(\mathcal{W})^{\\#_{Y}}\binom{X_{1},\ldots,X_{d}}{Y_{1},\ldots,Y_{d}}\rho_{\mathcal{B}}(\mathcal{W})^{\\#_{Y}}\binom{X_{d+1},\ldots,X_{d+d^{\prime}}}{Y_{d+1},\ldots,Y_{d+d^{\prime}}}$
$\displaystyle\hskip
199.16928pt=\sum_{\sigma\in\shuffle(d,d^{\prime})}\rho_{\mathcal{B}}(\mathcal{W})^{\\#_{Y}}\binom{X_{\sigma^{-1}(1)},\ldots,X_{\sigma^{-1}(d+d^{\prime})}}{Y_{\sigma^{-1}(1)},\ldots,Y_{\sigma^{-1}(d+d^{\prime})}}$

Moreover, for every entry
$\binom{X_{j}}{\underline{Y_{j}}+\underline{Y_{j^{\prime}}}}$ in the lower
depth terms coming from the third line in the recursive expression of
$\ast_{b}$ given in Theorem 5.1, the predecessor in the lower row is given by
$\underline{Y_{j-1}}+\underline{Y_{j^{\prime}-1}}$ (with
$\underline{Y_{j-1}}=0$ for $j=1,d+1$). Therefore, the terms of lower depth
equals the ones in the stuffle product formula on generating series of words
given in Theorem 4.1. Altogether, this means the product

$\rho_{\mathcal{B}}(\mathcal{W})^{\\#_{Y}}\binom{X_{1},\ldots,X_{d}}{Y_{1},\ldots,Y_{d}}\rho_{\mathcal{B}}(\mathcal{W})^{\\#_{Y}}\binom{X_{d+1},\ldots,X_{d+d^{\prime}}}{Y_{d+1},\ldots,Y_{d+d^{\prime}}}$

has an expression via the same recursive formula as for the stuffle product
obtained in Theorem 4.1. Thus, $\varphi_{\\#}$ is an algebra morphism. ∎

######  Corollary 7.5.

A bimould $M=(M_{d})_{d\geq 0}$ is b-symmetril if and only if the bimould
$M^{\\#_{Y}}$ is symmetril.

###### Proof.

This is a direct consequence of Theorem 7.4 and the definition of symmetrility
and b-symmetrility (compare to Definition 4.2, 5.2). ∎

The algebra morphism
$\varphi_{\\#}:\mathbb{Q}\langle\mathcal{Y}^{\operatorname{bi}}\rangle\to\mathbb{Q}\langle\mathcal{B}\rangle^{0}$
given in Theorem 7.4 is compatible with the coproducts
$\Delta_{\operatorname{dec}}$ and $\Delta_{\operatorname{dec}}^{0}$.

######  Proposition 7.6.

For all $d\geq 1$, we have

$(\varphi_{\\#}\otimes\varphi_{\\#})\circ\Delta_{\operatorname{dec}}=\Delta_{\operatorname{dec}}^{0}\circ\varphi_{\\#}.$

###### Proof.

We will prove this on the level of generating series of words by using the
formulas for $\Delta_{\operatorname{dec}}$ and
$\Delta_{\operatorname{dec}}^{0}$ given in Proposition 4.4 and 6.5. This
allows us to straight-forwardly compute
$\displaystyle(\\#_{Y}\otimes\\#_{Y})\circ\Delta_{\operatorname{dec}}\rho_{\mathcal{Y}^{\operatorname{bi}}}(\mathcal{W})\binom{X_{1},\ldots,X_{d}}{Y_{1},\ldots,Y_{d}}$
$\displaystyle=\sum_{i=0}^{d}\rho_{\mathcal{Y}^{\operatorname{bi}}}(\mathcal{W})^{\\#_{Y}}\binom{X_{1},\ldots,X_{i}}{Y_{1},\ldots,Y_{i}}\otimes\rho_{\mathcal{Y}^{\operatorname{bi}}}(\mathcal{W})^{\\#_{Y}}\binom{X_{i+1},\ldots,X_{d}}{Y_{i+1},\ldots,Y_{d}}$
$\displaystyle=\sum_{i=0}^{d}\rho_{\mathcal{Y}^{\operatorname{bi}}}(\mathcal{W})\binom{X_{1},\ldots,X_{i}}{Y_{1},Y_{1}+Y_{2},\ldots,Y_{1}+\dots+Y_{i}}\otimes\rho_{\mathcal{Y}^{\operatorname{bi}}}(\mathcal{W})\binom{X_{i+1},\ldots,X_{d}}{Y_{i+1},Y_{i+1}+Y_{i+2},\ldots,Y_{i+1}+\dots+Y_{d}}.$
On the other hand, we have
$\displaystyle\Delta_{\operatorname{dec}}^{0}\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})^{\\#_{Y}}\binom{X_{1},\ldots,X_{d}}{Y_{1},\ldots,Y_{d}}=\Delta_{\operatorname{dec}}^{0}\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{1},\ldots,X_{d}}{Y_{1},Y_{1}+Y_{2},\ldots,Y_{1}+\dots+Y_{d}}$
$\displaystyle=\sum_{i=0}^{d}\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{1},\ldots,X_{i}}{Y_{1},Y_{1}+Y_{2},\ldots,Y_{1}+\dots+Y_{i}}\otimes\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{i+1},\ldots,X_{d}}{Y_{i+1},Y_{i+1}+Y_{i+2},\ldots,Y_{i+1}+\dots+Y_{d}}.$
By definition of the map $\varphi_{\\#}$ (Definition 7.2) we get the claimed
formula. ∎

There are two involutions defined on $\mathbb{Q}\langle\mathcal{B}\rangle^{0}$
and $\mathbb{Q}\langle\mathcal{Y}^{\operatorname{bi}}\rangle$, which play an
important role in the theory of multiple q-zeta values and which are
compatible with the morphism $\varphi_{\\#}$.

######  Definition 7.7.

Let
$\tau:\mathbb{Q}\langle\mathcal{B}\rangle^{0}\to\mathbb{Q}\langle\mathcal{B}\rangle^{0}$
be the involution given by $\tau(\mathbf{1})=\mathbf{1}$ and

$\tau(b_{k_{1}}b_{0}^{m_{1}}\dots
b_{k_{d}}b_{0}^{m_{d}})=b_{m_{d}+1}b_{0}^{k_{d}-1}\dots
b_{m_{1}+1}b_{0}^{k_{1}-1}$

for all $k_{1},\ldots,k_{d}\geq 1,\ m_{1},\ldots,m_{d}\geq 0$.

Extend the involution $\tau$ by
$\mathbb{Q}[X_{1},Y_{1},X_{2},Y_{2},\ldots]$-linearity to
$\mathbb{Q}\langle\mathcal{B}\rangle^{0}\llbracket
X_{1},Y_{1},X_{2},Y_{2},\ldots\rrbracket$, then we obtain directly for each
$d\geq 0$ that

$\tau\Bigg{(}\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{1},\ldots,X_{d}}{Y_{1},\ldots,Y_{d}}\Bigg{)}=\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{Y_{d},\ldots,Y_{1}}{X_{d},\ldots,X_{1}}.$

This motivates the following definition for bimoulds.

######  Definition 7.8.

For a bimould $M=(M_{d})_{d\geq 0}$, define the bimould
$\tau(M)=(\tau(M)_{d})_{d\geq 0}$ by

$\displaystyle\operatorname{\tau}(M)_{d}\binom{X_{1},\ldots,X_{d}}{Y_{1},\ldots,Y_{d}}$
$\displaystyle=M_{d}\binom{Y_{d},\ldots,Y_{1}}{X_{d},\ldots,X_{1}}.$

We call a bimould $M$ _$\tau$ -invariant_ if $\tau(M)=M$.

The second involution defined on the algebra
$\mathbb{Q}\langle\mathcal{Y}^{\operatorname{bi}}\rangle$ is originally given
in terms of bimoulds, since its expression is much easier in this case.

######  Definition 7.9.

For a bimould $M=(M_{d})_{d\geq 0}$, define the bimould
$\operatorname{swap}(M)=(\operatorname{swap}(M)_{d})_{d\geq 0}$ as

$\operatorname{swap}(M)_{d}\binom{X_{1},\ldots,X_{d}}{Y_{1},\ldots,Y_{d}}=M_{d}\binom{Y_{1}+\dots+Y_{d},Y_{1}+\dots+Y_{d-1},\ldots,Y_{1}}{X_{d},X_{d-1}-X_{d},\ldots,X_{1}-X_{2}}.$

A bimould $M$ is called _swap invariant_ if $\operatorname{swap}(M)=M$.
Define the involution
$\operatorname{swap}:\mathbb{Q}\langle\mathcal{Y}^{\operatorname{bi}}\rangle\to\mathbb{Q}\langle\mathcal{Y}^{\operatorname{bi}}\rangle$
implicitly by

$(\operatorname{swap}\otimes\rho_{\mathcal{Y}^{\operatorname{bi}}})(\mathcal{W})_{d}\binom{X_{1},\ldots,X_{d}}{Y_{1},\ldots,Y_{d}}=\operatorname{swap}(\rho_{\mathcal{Y}^{\operatorname{bi}}}(\mathcal{W}))_{d}\binom{X_{1},\ldots,X_{d}}{Y_{1},\ldots,Y_{d}},$

i.e., the coefficient of $X_{1}^{k_{1}-1}\frac{Y_{1}^{m_{1}}}{m_{1}!}\dots
X_{d}^{k_{d}-1}\frac{Y_{d}^{m_{d}}}{m_{d}!}$ in
$\operatorname{swap}(\rho_{\mathcal{Y}^{\operatorname{bi}}}(\mathcal{W}))_{d}$
is the image of the word $y_{k_{1},m_{1}}\dots y_{k_{d},m_{d}}$ under the map
$\operatorname{swap}$.

For example, we obtain

$\displaystyle\operatorname{swap}(y_{k_{1},m_{1}})$
$\displaystyle=\frac{m_{1}!}{(k_{1}-1)!}y_{m_{1}+1,k_{1}-1},$ (7.3)
$\displaystyle\operatorname{swap}(y_{k_{1},m_{1}}y_{k_{2},m_{2}})$
$\displaystyle=\sum_{u=0}^{m_{1}}\sum_{v=0}^{k_{2}-1}\frac{(-1)^{v}}{u!v!}\frac{m_{1}!}{(k_{1}-1)!}\frac{(m_{2}+u)!}{(k_{2}-1-v)!}y_{m_{2}+1+u,k_{2}-1-v}y_{m_{1}+1-u,k_{1}-1+v}.$

In higher depths, it is hard to give an explicit formula for the involution
$\operatorname{swap}$ on the algebra
$\mathbb{Q}\langle\mathcal{Y}^{\operatorname{bi}}\rangle$, see for example
[BI22, Remark 3.14].

######  Theorem 7.10.

The map

$\varphi_{\\#}:(\mathbb{Q}\langle\mathcal{Y}^{\operatorname{bi}}\rangle,\ast,\Delta_{\operatorname{dec}})\to(\mathbb{Q}\langle\mathcal{B}\rangle^{0},\ast_{b},\Delta_{\operatorname{dec}}^{0})$

is an isomorphism of weight-graded Hopf algebras satisfying
$\varphi_{\\#}\circ\operatorname{swap}=\tau\circ\varphi_{\\#}$.

###### Proof.

The map $\varphi_{\\#}$ is an isomorphism of weight-graded Hopf algebras by
Theorem 7.4 and Proposition 7.6. Moreover, one verifies directly on the level
of generating series of words

$\displaystyle\\#_{Y}\circ\operatorname{swap}\rho_{\mathcal{Y}^{\operatorname{bi}}}(\mathcal{W})\binom{X_{1},\ldots,X_{d}}{Y_{1},\ldots,Y_{d}}$
$\displaystyle=\rho_{\mathcal{Y}^{\operatorname{bi}}}(\mathcal{W})^{\\#_{Y}}\binom{Y_{1}+\dots+Y_{d},Y_{1}+\dots+Y_{d-1},\ldots,Y_{1}}{X_{d},X_{d-1}-X_{d},\ldots,X_{1}-X_{2}}$
$\displaystyle=\rho_{\mathcal{Y}^{\operatorname{bi}}}(\mathcal{W})\binom{Y_{1}+\dots+Y_{d},Y_{1}+\dots+Y_{d-1},\ldots,Y_{1}}{X_{d},X_{d-1},\ldots,X_{1}},$

and

$\displaystyle\tau\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})^{\\#_{Y}}\binom{X_{1},\ldots,X_{d}}{Y_{1},\ldots,Y_{d}}$
$\displaystyle=\tau\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{X_{1},\ldots,X_{d}}{Y_{1},Y_{1}+Y_{2},\ldots,Y_{1}+\dots+Y_{d}}$
$\displaystyle=\rho_{\mathcal{B}}^{0}(\mathcal{W}^{0})\binom{Y_{1}+\dots+Y_{d},Y_{1}+\dots+Y_{d-1},\ldots,Y_{1}}{X_{d},X_{d-1},\ldots,X_{1}}.$

By definition of the map $\varphi_{\\#}$ (Definition 7.2), we deduce that
$\varphi_{\\#}\circ\operatorname{swap}=\tau\circ\varphi_{\\#}$. ∎

An immediate consequence of Theorem 7.10 is the following.

######  Corollary 7.11.

A bimould $M=(M_{d})_{d\geq 0}$ is $\tau$-invariant if and only if the bimould
$M^{\\#_{Y}}$ is swap invariant.

##  8\. The algebra of multiple $\operatorname{q}$-zeta values

We introduce the algebra of multiple q-zeta values $\mathcal{Z}_{q}$ and
explain its relations to multiple zeta values and polynomial functions on
partitions. As an application of the previous general results, we will present
a particular nice spanning set for $\mathcal{Z}_{q}$, the balanced multiple
q-zeta values.

######  Definition 8.1.

([BK20]) To integers $s_{1}\geq 1,\ s_{2},\ldots,s_{l}\geq 0$ and polynomials
$R_{1}\in t\mathbb{Q}[t],\\\ R_{2},\ldots,R_{l}\in$ $\mathbb{Q}[t]$, associate
the _generic multiple q-zeta value_

$\zeta_{q}(s_{1},...,s_{l};R_{1},...,R_{l})=\sum_{n_{1}>\dots>n_{l}>0}\frac{R_{1}(q^{n_{1}})}{(1-q^{n_{1}})^{s_{1}}}\cdots\frac{R_{l}(q^{n_{l}})}{(1-q^{n_{l}})^{s_{l}}}\in\mathbb{Q}\llbracket
q\rrbracket.$

The assumptions $s_{1}\geq 1$ and $R_{1}\in t\mathbb{Q}[t]$ are necessary for
convergence.

######  Proposition 8.2.

([BK20, p. 6]) For integers $s_{1}\geq 2,\ s_{2},\ldots,s_{l}\geq 1$ and
polynomials $R_{1}\in t\mathbb{Q}[t],\ R_{2},\ldots,R_{l}\in$ $\mathbb{Q}[t]$,
one has

$\lim_{q\to
1}(1-q)^{s_{1}+\dots+s_{l}}\zeta_{q}(s_{1},...,s_{l};R_{1},...,R_{l})=R_{1}(1)\cdots
R_{l}(1)\zeta(s_{1},...,s_{l}).$

∎

Following H. Bachmann and U. Kühn ([BK20]), we consider the following kind of
multiple q-zeta values.

######  Definition 8.3.

Define the $\mathbb{Q}$-vector space

$\mathcal{Z}_{q}=\operatorname{span}_{\mathbb{Q}}\\!\big{\\{}\zeta_{q}(s_{1},...,s_{l};R_{1},...,R_{l})\
\big{|}\ l\geq 0,s_{1}\geq 1,\ s_{2},...,s_{l}\geq 0,\deg(R_{j})\leq
s_{j}\big{\\}},$

where we set $\zeta_{q}(\emptyset;\emptyset)=1$.

The space $\mathcal{Z}_{q}$ contains all models for multiple q-zeta values
studied in the literature, so this can be seen as a model-free approach to
them. For $\zeta_{q}(s_{1};R_{1}),\zeta_{q}(s_{2};R_{2})\in\mathcal{Z}_{q}$,
the usual power series multiplication reads

$\zeta_{q}(s_{1};R_{1})\cdot\zeta_{q}(s_{2};R_{2})=\zeta_{q}(s_{1},s_{2};R_{1},R_{2})+\zeta_{q}(s_{2},s_{1};R_{2},R_{1})+\zeta_{q}(s_{1}+s_{2};R_{1}R_{2}).$

Since $\deg(R_{1}R_{2})\leq s_{1}+s_{2}$, the product is also an element in
$\mathcal{Z}_{q}$. Similar computations for arbitrary multi indices show that
$\mathcal{Z}_{q}$ is an associative, commutative algebra. Moreover, the
algebra $\mathcal{Z}_{q}$ contains the algebra
$\widetilde{\mathcal{M}}^{\mathbb{Q}}(\operatorname{SL}_{2}(\mathbb{Z}))$ of
quasi-modular forms with rational coefficients (as introduced in [KZ95]) and
is closed under the well-known derivation
$q\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}q}$.

######  Remark 8.4.

The additional assumption on the degree of the polynomials $R_{j}$ in
Definition 8.3 can be justified by its relations to polynomial functions on
partitions. More precisely, the original definition of the space
$\mathcal{Z}_{q}$ was given by H. Bachmann as the span of the bi-brackets
([Ba19]), which can be seen as generating series of monomial functions on
partitions. In [BI22, cf (1.6)] it is shown that the space $\mathcal{Z}_{q}$
is exactly the image of the polynomial functions on partitions under the
q-bracket.
Let $\lambda=(1^{m_{1}}2^{m_{2}}3^{m_{3}}\dots)$ be a partition of some
natural number $N$ of length $d$, i.e., the multiplicities
$m_{i}\in\mathbb{Z}_{\geq 0}$ are nonzero only for finitely many indices
$i_{1},\ldots,i_{d}$ and one has $\sum_{i\geq 1}m_{i}i=N$. A polynomial
$f\in\mathbb{Q}[X_{1},\ldots,X_{d},Y_{1},\ldots,Y_{d}]$ can be evaluated at
the partition $\lambda$ by

$f(\lambda)=f(i_{1},\ldots,i_{d},m_{i_{1}},\ldots,m_{i_{d}}).$

Denote by $\mathcal{P}(N,d)$ the set of all partitions of $N$ of length $d$.
Then we associate to a polynomial
$f\in\mathbb{Q}[X_{1},\ldots,X_{d},Y_{1},\ldots,Y_{d}]$ the generating series

$\operatorname{Gen}_{f}(q)=\sum_{N\geq
1}\left(\sum_{\lambda\in\mathcal{P}(N,d)}f(\lambda)\right)q^{N}.$

Whenever $f\in\mathbb{Q}[X_{1},\ldots,X_{d},Y_{1},\ldots,Y_{d}]$ is chosen to
be a monomial, then $\operatorname{Gen}_{f}(q)$ is equal to a bi-bracket of
depth $d$, more details are given in [Bri21, Theorem 1.3]. Therefore, as a
consequence of [BK20, Theorem 2.3] we obtain that the space $\mathcal{Z}_{q}$
is spanned by the generating series $\operatorname{Gen}_{f}(q)$ with
$f\in\mathbb{Q}[X_{1},\ldots,X_{d},Y_{1},\ldots,Y_{d}]$, $d\geq 0$. This
indicates that the elements in $\mathcal{Z}_{q}$ should be invariant under
some involution corresponding to conjugation of partitions.

##  9\. Combinatorial bi-multiple Eisenstein series

We present the combinatorial bi-multiple Eisenstein series constructed in
[BB22]. They form a spanning set for the space $\mathcal{Z}_{q}$, so they are
a particular model for multiple q-zeta values. Combinatorial bi-multiple
Eisenstein series satisfy a weight-graded product formula and are invariant
under a weight-homogeneous involution. Therefore, they should induce a grading
on $\mathcal{Z}_{q}$, which extends the weight-grading of the algebra
$\widetilde{\mathcal{M}}^{\mathbb{Q}}(\operatorname{SL}_{2}(\mathbb{Z}))$ of
quasi-modular forms with rational coefficients. Their construction is inspired
by the Fourier expansion of multiple Eisenstein series ([Ba20, Theorem 1.4]).
Thus, the combinatorial bi-multiple Eisenstein series give a natural
connection between the space $\mathcal{Z}_{q}$ and multiple Eisenstein series.
In particular, they should give a description of all relations between
multiple Eisenstein series. We will recall the construction of the
combinatorial bi-multiple Eisenstein series as given in [BB22].

######  Definition 9.1.

By the work of G. Racinet ([Rac00]) and also the combination of the works of
V. G. Drinfeld ([Dr91]) and H. Furusho ([Fu11]) there exists a rational
solution to the extended double shuffle equations

$\beta(k_{1},\ldots,k_{d})\in\mathbb{Q},\qquad
k_{1},\ldots,k_{d}\in\mathbb{Z}_{\geq 1},\ k_{1}>1,$

such that

$\displaystyle\beta(k)=-\frac{B_{k}}{2k!},\ k\text{ even},\qquad\beta(k)=0,\
k\text{ odd}.$ (9.1)

Denote by $\beta_{\ast}(k_{1},\ldots,k_{d}),\ k_{1},\ldots,k_{d}\geq 1$, the
corresponding stuffle regularized elements and define for all $d\geq 1$

$\displaystyle\mathfrak{b}_{d}(X_{1},\ldots,X_{d})=\sum_{k_{1},\ldots,k_{d}\geq
1}\beta_{\ast}(k_{1},\ldots,k_{d})X_{1}^{k_{1}-1}\dots X_{d}^{k_{d}-1}.$ (9.2)

Moreover set $\mathfrak{b}_{0}=1$, then
$\mathfrak{b}=(\mathfrak{b}_{d})_{d\geq 0}$ is a mould with coefficients in
$\mathbb{Q}$.

In general, a solution to the extended double shuffle equations is not unique.
In the following, we will fix the mould $\mathfrak{b}$ and the whole
construction of the combinatorial bi-multiple Eisenstein series will depend on
this choice.

######  Definition 9.2.

Define the bimould $\mathfrak{b}=(\mathfrak{b}_{d})_{d\geq 0}$ by
$\mathfrak{b}_{0}=1$ and for $d\geq 1$ by

$\displaystyle\mathfrak{b}_{d}\binom{X_{1},\dots,X_{d}}{Y_{1},\dots,Y_{d}}=\sum_{0\leq
i\leq j\leq
d}\gamma_{i}\mathfrak{b}_{j-i}(Y_{1}+\dots+Y_{j-i},\dots,Y_{1}+Y_{2},Y_{1})\mathfrak{b}_{d-j}(X_{j+1},\dots,X_{d}),$

where the coefficients $\gamma_{i}$ are defined by

$\displaystyle\sum_{i\geq 0}\gamma_{i}T^{i}=\exp\left(\sum_{n\geq
2}\frac{(-1)^{n+1}}{n}\frac{B_{n}}{2n!}T^{n}\right).$

Independent of the shape of the mould $\mathfrak{b}$, this construction will
always yield a swap invariant bimould. Since the coefficients of the mould
$\mathfrak{b}$ satisfy the extended double shuffle relations, we obtain that
the bimould $\mathfrak{b}$ is also symmetril.

######  Definition 9.3.

Define the bimould
$\widetilde{\mathfrak{b}}=(\widetilde{\mathfrak{b}}_{d})_{d\geq 0}$ by
$\widetilde{\mathfrak{b}}_{0}=1$ and

$\displaystyle\widetilde{\mathfrak{b}}_{d}\binom{X_{1},\ldots,X_{d}}{Y_{1},\ldots,Y_{d}}=\sum_{i=0}^{d}\frac{(-1)^{i}}{2^{i}i!}\mathfrak{b}_{d-i}\binom{X_{i+1},\ldots,X_{d}}{-Y_{1},\ldots,-Y_{d-i}},\qquad
d\geq 1.$

For each $u\geq 1$, let $\mathfrak{L}^{(u)}=(\mathfrak{L}^{(u)}_{d})_{d\geq
0}$ be the bimould given by $\mathfrak{L}^{(u)}_{0}=1$ and

$\displaystyle\mathfrak{L}_{d}^{(u)}\binom{X_{1},\ldots,X_{d}}{Y_{1},\ldots,Y_{d}}=\sum_{j=1}^{d}\mathfrak{b}_{j-1}\binom{X_{1}-X_{j},\ldots,X_{j-1}-X_{j}}{Y_{1},\ldots,Y_{j-1}}L_{u}\binom{X_{j}}{Y_{1}+\dots+Y_{d}}$
$\displaystyle\hskip
284.52756pt\cdot\widetilde{\mathfrak{b}}_{d-j}\binom{X_{d}-X_{j},\ldots,X_{j+1}-X_{j}}{Y_{d},\ldots,Y_{j+1}},$

where the power series $L_{u}\binom{X}{Y}$ is defined by

$L_{u}\binom{X}{Y}=\frac{\exp(X+uY)q^{u}}{1-\exp(X)q^{u}},\qquad u\geq 1.$

It can be shown ([BB22, Lemma 6.20]) that the bimould $\mathfrak{L}^{(u)}$ is
symmetril, the proof uses the additional conditions on the depth $1$ terms of
$\mathfrak{b}$ given in (9.1).

######  Definition 9.4.

Define the bimould $\mathfrak{g}^{\ast}=(\mathfrak{g}^{\ast}_{d})_{d\geq 0}$
by $\mathfrak{g}^{\ast}_{0}=1$ and

$\displaystyle\mathfrak{g}^{\ast}_{d}\binom{X_{1},\ldots,X_{d}}{Y_{1},\ldots,Y_{d}}=\sum_{\begin{subarray}{c}1\leq
j\leq d\\\ 0=d_{0}<d_{1}<\dots<d_{j-1}<d_{j}=d\\\
u_{1}>\dots>u_{j}>0\end{subarray}}\prod_{i=1}^{j}\mathfrak{L}^{(u_{i})}_{d_{i}-d_{i-1}}\binom{X_{d_{i-1}+1},\ldots,X_{d_{i}}}{Y_{d_{i-1}+1},\ldots,Y_{d_{i}}}.$

Since the bimould $\mathfrak{L}^{(u)}$ is symmetril, also
$\mathfrak{g}^{\ast}$ is symmetril. This implication is independent of the
explicit shape of $\mathfrak{L}^{(u)}$.

######  Definition 9.5.

Let $\mathfrak{G}=(\mathfrak{G}_{d})_{d\geq 0}$ be the mould product of
$\mathfrak{g}^{\ast}$ and $\mathfrak{b}$, i.e., one has $\mathfrak{G}_{0}=1$
and for $d\geq 1$

$\displaystyle\mathfrak{G}_{d}\binom{X_{1},\ldots,X_{d}}{Y_{1},\ldots,Y_{d}}=\sum_{i=0}^{d}\mathfrak{g}_{i}^{\ast}\binom{X_{1},\ldots,X_{i}}{Y_{1},\ldots,Y_{i}}\mathfrak{b}_{d-i}\binom{X_{i+1},\ldots,X_{d}}{Y_{i+1},\ldots,Y_{d}}.$

The main result of [BB22] was the following.

######  Theorem 9.6.

[BB22, Theorem 6.5.] The bimould $\mathfrak{G}$ is symmetril and swap
invariant. ∎

As a product of symmetril bimoulds, $\mathfrak{G}$ is symmetril. To obtain the
swap invariance, one decomposes $\mathfrak{G}$ into a sum of bimoulds, for
which the swap invariance can be shown.

######  Definition 9.7.

For $k_{1},\dots,k_{d}\geq 1$, $m_{1},\dots,m_{d}\geq 0$ define the
_combinatorial bi-multiple Eisenstein series_
$G\binom{k_{1},\ldots,k_{d}}{m_{1},\ldots,m_{d}}$ to be the (normalized)
coefficients of the bimould $\mathfrak{G}$,

$\displaystyle\mathfrak{G}_{d}\binom{X_{1},\ldots,X_{d}}{Y_{1},\ldots,Y_{d}}=\sum_{\begin{subarray}{c}k_{1},\dots,k_{d}\geq
1\\\ m_{1},\dots,m_{d}\geq
0\end{subarray}}G\binom{k_{1},\ldots,k_{d}}{m_{1},\ldots,m_{d}}X_{1}^{k_{1}-1}\frac{Y_{1}^{m_{1}}}{m_{1}!}\dots
X_{d}^{k_{d}-1}\frac{Y_{d}^{m_{d}}}{m_{d}!}.$

We refer to the number $k_{1}+\dots+k_{d}+m_{1}+\dots+m_{d}$ as the weight of
$G\binom{k_{1},\ldots,k_{d}}{m_{1},\ldots,m_{d}}$ and to the number $d$ as its
depth. Moreover, the elements

$G(k_{1},\dots,k_{d})=G\binom{k_{1},\ldots,k_{d}}{0,\ldots,0},\qquad
k_{1},\ldots,k_{d}\geq 1,$

are called the _combinatorial multiple Eisenstein series_.

######  Example 9.8.

In depth $1$, one obtains

$\mathfrak{G}_{1}\binom{X_{1}}{Y_{1}}=\mathfrak{b}_{1}\binom{X_{1}}{Y_{1}}+\sum_{u>0}L_{u}\binom{X_{1}}{Y_{1}}$

and thus for $k\geq 1,\ m\geq 0$

$G\binom{k}{m}=-\delta_{m,0}\frac{B_{k}}{2k!}-\delta_{k,1}\frac{B_{m+1}}{2(m+1)}+\frac{1}{(k-1)!}\sum_{u,v>0}u^{m}v^{k-1}q^{uv}.$

In particular, the combinatorial Eisenstein series $G(k)$, $k\geq 2$ even, are
exactly the classical Eisenstein series of weight $k$ with rational
coefficients (expressed in their Fourier expansion). The combinatorial bi-
Eisenstein series $G\binom{k}{m}$, $k+m\geq 2$ even, is essentially the $m$-th
derivative of the classical Eisenstein series $G(k)$ and hence is also
contained in the algebra
$\widetilde{\mathcal{M}}^{\mathbb{Q}}(\operatorname{SL}_{2}(\mathbb{Z}))$ of
quasi-modular forms with rational coefficients.

######  Proposition 9.9.

([BB22, Proposition 6.15.]) The combinatorial bi-multiple Eisenstein series
form a spanning set of $\mathcal{Z}_{q}$. ∎

Theorem 9.6, the definition of the symmetrility and swap (compare to
Definition 4.2, 7.9), and Proposition 9.9 imply the following.

######  Corollary 9.10.

There is a surjective, swap invariant algebra morphism

$\displaystyle
G:(\mathbb{Q}\langle\mathcal{Y}^{\operatorname{bi}}\rangle,\ast)$
$\displaystyle\to\mathcal{Z}_{q},$ $\displaystyle y_{k_{1},m_{1}}\dots
y_{k_{d},m_{d}}$ $\displaystyle\mapsto
G\binom{k_{1},\ldots,k_{d}}{m_{1},\ldots,m_{d}}.$

######  Conjecture 9.11.

([BB22, Remark 6.11.]) All algebraic relations in $\mathcal{Z}_{q}$ are a
consequence of the stuffle product formula and the swap invariance of the
combinatorial bi-multiple Eisenstein series.

If Conjecture 9.11 holds, then the algebra $\mathcal{Z}_{q}$ is graded by
weight

$\displaystyle\mathcal{Z}_{q}=\bigoplus_{w\geq 0}\mathcal{Z}_{q}^{(w)}.$ (9.3)

Here $\mathcal{Z}_{q}^{(w)}$ denotes the subspace of $\mathcal{Z}_{q}$ spanned
by all combinatorial bi-multiple Eisenstein series of weight $w$. This follows
immediately from the observation that the stuffle product and also the swap
operator are homogeneous in weight.

##  10\. Balanced multiple $\operatorname{q}$-zeta values

We will apply the results from Section 7 to obtain a new spanning set of the
algebra $\mathcal{Z}_{q}$. They will satisfy very explicit and simple
relations, which are homogeneous in weight. Thus, this new spanning set should
induce a weight-grading on $\mathcal{Z}_{q}$.

######  Definition 10.1.

Define the bimould $\mathfrak{B}=(\mathfrak{B}_{d})_{d\geq 0}$ with
coefficients in $\mathcal{Z}_{q}$ by $\mathfrak{B}_{0}=1$ and for $d\geq 1$ by

$\mathfrak{B}_{d}\binom{X_{1},\ldots,X_{d}}{Y_{1},\ldots,Y_{d}}=\mathfrak{G}^{\\#_{Y}^{-1}}_{d}\binom{X_{1},\ldots,X_{d}}{Y_{1},\ldots,Y_{d}},$

where $\mathfrak{G}$ is the bimould of generating series of the combinatorial
bi-multiple Eisenstein series (Definition 9.5). The _balanced multiple q-zeta
values_ $\zeta_{q}(s_{1},\ldots,s_{l})$, $s_{1}\geq 1$,
$s_{2},\ldots,s_{l}\geq 0$, are the coefficients of the bimoulds
$\mathfrak{B}$,

$\displaystyle\mathfrak{B}_{d}\binom{X_{1},\ldots,X_{d}}{Y_{1},\ldots,Y_{d}}=\sum_{\begin{subarray}{c}k_{1},\dots,k_{d}\geq
1\\\ m_{1},\dots,m_{d}\geq
0\end{subarray}}\zeta_{q}(k_{1},\\{0\\}^{m_{1}},\ldots,k_{d},\\{0\\}^{m_{d}})X_{1}^{k_{1}-1}Y_{1}^{m_{1}}\dots
X_{d}^{k_{d}-1}Y_{d}^{m_{d}},\quad d\geq 1.$

Equivalently, the balanced multiple q-zeta values can be defined as

$\zeta_{q}(s_{1},\ldots,s_{l})=G(\varphi_{\\#}^{-1}(b_{s_{1}}\dots
b_{s_{l}})),$

where
$G:\mathbb{Q}\langle\mathcal{Y}^{\operatorname{bi}}\rangle\to\mathcal{Z}_{q}$
denotes the algebra morphism induced by the combinatorial bi-multiple
Eisenstein series given in Corollary 9.10.

######  Remark 10.2.

The definition of the balanced multiple q-zeta values has many similarities
with Zudilin’s definition of the multiple q-zeta brackets in terms of the bi-
brackets ([Zu15, eq (8)]).

Since the bimould $\mathfrak{G}$ of generating series of the combinatorial bi-
multiple Eisenstein series is symmetril and swap invariant (Theorem 9.6), an
immediate consequence of Corollary 7.5 and 7.11 is the following.

######  Proposition 10.3.

The bimould $\mathfrak{B}$ is b-symmetril and $\tau$-invariant.

These symmetries of the bimould $\mathfrak{B}$ yield the following properties
of balanced multiple q-zeta values.

######  Theorem 10.4.

There is a $\tau$-invariant, surjective algebra morphism

$\displaystyle\zeta_{q}:(\mathbb{Q}\langle\mathcal{B}\rangle^{0},\ast_{b})$
$\displaystyle\to\mathcal{Z}_{q},$ $\displaystyle b_{s_{1}}\dots b_{s_{l}}$
$\displaystyle\mapsto\zeta_{q}(s_{1},\ldots,s_{l}).$

###### Proof.

This is an immediate consequence of the b-symmetrility and $\tau$-invariance
of the bimould $\mathfrak{B}$ (see Definition 5.2, 7.8) together with the
observation that $\\#_{Y}$ is a bijection. ∎

As a reformulation of Conjecture 9.11, we expect the following.

######  Conjecture 10.5.

All algebraic relations in $\mathcal{Z}_{q}$ are a consequence of the balanced
quasi-shuffle product formula and the $\tau$-invariance of the balanced
multiple q-zeta values.

Observe that the balanced quasi-shuffle product, as well as the
$\tau$-invariance, are both completely explicit on the level of words of multi
indices and easy to compute. If Conjecture 10.5 holds, then the algebra
$\mathcal{Z}_{q}$ is graded by weight

$\mathcal{Z}_{q}=\bigoplus_{w\geq 0}\mathcal{Z}_{q}^{(w)},$

where $\mathcal{Z}_{q}^{(w)}$ is the subspace of $\mathcal{Z}_{q}$ spanned by
all balanced multiple q-zeta values of weight $w$. This conjectural grading
coincides with the one of the combinatorial bi-multiple Eisenstein series
given in (9.3).
It was conjectured in [Ba19, Conjecture 4.3] in a slightly different setup
that the space $\mathcal{Z}_{q}$ is spanned by the elements
$\zeta_{q}(k_{1},\ldots,k_{d})$, $k_{1},\ldots,k_{d}\geq 1$. Some partial
results towards this conjecture are obtained in [Bu23, Section 6], [Ba19,
Proposition 4.4, 5.9] and [Vl20, Theorem 5.3] and a general proof is announced
in [BBI23].

######  Example 10.6.

1\. In depth $1$ the balanced multiple q-zeta values coincide with the
combinatorial bi-multiple Eisenstein series up to multiplication with certain
factorials. Thus, we deduce from Example 9.8 that for all $k\geq 1,\ m\geq 0$

$\zeta_{q}(k,\\{0\\}^{m})=-\delta_{m,0}\frac{B_{k}}{2k!}-\delta_{k,1}\frac{B_{m+1}}{2(m+1)!}+\frac{1}{(k-1)!m!}\sum_{u,v>0}u^{m}v^{k-1}q^{uv}.$

So, the element $\zeta_{q}(k,\\{0\\}^{m})$ is essentially equal to the $m$-th
derivative of the Eisenstein series $G_{k}$. If $k_{1},\ldots,k_{d}\geq 1$,
then the balanced multiple q-zeta value $\zeta_{q}(k_{1},\ldots,k_{d})$ equals
the combinatorial multiple Eisenstein series $G(k_{1},\ldots,k_{d})$. In
particular, the balanced multiple q-zeta values give a very natural extension
and explicit description of conjectural all relations between multiple
Eisenstein series.
2\. Recall that the mould $\mathfrak{b}$ in depth $2$ is given by
$\mathfrak{b}(X_{1},X_{2})=\sum_{k_{1},k_{2}\geq
1}\beta_{\ast}(k_{1},k_{2})X_{1}^{k_{1}-1}X_{2}^{k_{2}-1}$ (compare to (9.2)).
Direct calculations show that

$\displaystyle\zeta_{q}(2,3)$
$\displaystyle=\beta_{\ast}(2,3)-\frac{1}{48}\sum_{u,v>0}v^{2}q^{uv}+\frac{1}{2}\sum_{\begin{subarray}{c}u_{1}>u_{2}>0\\\
v_{1},v_{2}>0\end{subarray}}v_{1}v_{2}^{2}q^{u_{1}v_{1}+u_{2}v_{2}},$
$\displaystyle\zeta(2,0,3)$
$\displaystyle=\frac{1}{2}\sum_{\begin{subarray}{c}u_{1}>u_{2}>0\\\
v_{1},v_{2}>0\end{subarray}}u_{1}v_{1}v_{2}^{2}q^{u_{1}v_{1}+u_{2}v_{2}}-\frac{1}{2}\sum_{\begin{subarray}{c}u_{1}>u_{2}>0\\\
v_{1},v_{2}>0\end{subarray}}u_{2}v_{1}v_{2}^{2}q^{u_{1}v_{1}+u_{2}v_{2}}.$

An explicit construction for the numbers $\beta_{\ast}(k_{1},k_{2})$ is given
in [GKZ06, Section 6], in this case one obtains $\beta_{\ast}(2,3)=0$. Further
constructions for these rational numbers are given in [Bro17], [Ec02].
3\. The quasi-modular forms are contained in the algebra $\mathcal{Z}_{q}$. In
particular, the modular discriminant $\Delta(q)=q\prod_{n\geq
1}(1-q^{n})^{24}$ is a $\mathbb{Q}$-linear combination of balanced multiple
q-zeta values. For example, using the exotic relation given in [Con22, Example
2.48 (ii)] we obtain

$\displaystyle\frac{1}{43200}\Delta(q)=$ $\displaystyle\
240\zeta_{q}(4,4,4)-63\zeta_{q}(9,3)+183\zeta_{q}(8,4)-\frac{675}{2}\zeta_{q}(7,5)+\frac{89}{2}\zeta_{q}(6,6)-378\zeta_{q}(5,7)$
$\displaystyle+183\zeta_{q}(4,8).$

##  11\. Further properties of the balanced multiple $\operatorname{q}$-zeta
values

Similar to the case of the Schlesinger-Zudilin multiple q-zeta values studied
by K. Ebrahimi-Fard, D. Manchon, and J. Singer ([Si15], [EMS16]), the balanced
multiple q-zeta values possess a description in terms of an alphabet with two
letters. We will compute the limits $q\to 1$ of a certain kind of balanced
multiple q-zeta values and obtain that those are elements in the algebra of
multiple zeta values. Finally, we will describe the derivation
$q\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}q}$ on the algebra
$\mathcal{Z}_{q}$ in terms of the balanced multiple q-zeta values.

######  Definition 11.1.

Let $\shuffle_{b}$ be the product on the non-commutative free algebra
$\mathbb{Q}\langle p,y\rangle$ recursively defined by
$\mathbf{1}\shuffle_{b}w=w\shuffle_{b}\mathbf{1}=w$ and

$\displaystyle(yu)\shuffle_{b}v$
$\displaystyle=u\shuffle_{b}(yv)=y(u\shuffle_{b}v),$
$\displaystyle(pu)\shuffle_{b}(pv)$
$\displaystyle=p(u\shuffle_{b}pv)+p(pu\shuffle_{b}v)+\begin{cases}p(u\shuffle_{b}v),&\text{
if }u=y\tilde{u}\text{ and }v=y\tilde{v},\\\ 0&\text{ else}\end{cases}$

for all $u,v,w\in\mathbb{Q}\langle p,y\rangle$.

The involution $\tau$ (Definition 7.7) can be also defined on the algebra
$\mathbb{Q}\langle p,y\rangle$.

######  Definition 11.2.

Let $\tau$ be the anti-automorphism on $\mathbb{Q}\langle p,y\rangle$ given by
$\tau(\mathbf{1})=\mathbf{1}$, $\tau(p)=y$, and $\tau(y)=p$, i.e., one has for
all $k_{1},\ldots,k_{d}\geq 1,\ m_{0},\ldots,m_{d}\geq 0$

$\displaystyle\tau(y^{m_{0}}p^{k_{1}}y^{m_{1}}\dots
p^{k_{d}}y^{m_{d}})=p^{m_{d}}y^{k_{d}}\dots p^{m_{1}}y^{k_{1}}p^{m_{0}}.$

The involution $\tau$ connects the balanced quasi-shuffle product $\ast_{b}$
and the product $\shuffle_{b}$. To make this precise, consider the canonical
embedding

$\displaystyle i:\mathbb{Q}\langle\mathcal{B}\rangle$
$\displaystyle\hookrightarrow\mathbb{Q}\langle p,y\rangle,$ $\displaystyle
b_{s_{1}}\dots b_{s_{l}}$ $\displaystyle\mapsto p^{s_{1}}y\dots p^{s_{l}}y.$

Observe that we have $\tau(i(w))=i(\tau(w))$ for all
$w\in\mathbb{Q}\langle\mathcal{B}\rangle^{0}$.

######  Proposition 11.3.

For all $u,v\in\mathbb{Q}\langle\mathcal{B}\rangle$, we have

$i(u\ast_{b}v)=\tau(\tau\circ i(u)\shuffle_{b}\tau\circ i(v)).$

###### Proof.

Let $u=b_{s_{1}}\dots b_{s_{l}}$ and $v=b_{r_{1}}\dots b_{r_{k}}$ be words in
$\mathbb{Q}\langle\mathcal{B}\rangle$ and $s_{l},r_{k}\geq 1$. Using the
recursive definition of $\ast_{b}$ from the right, we obtain

$\displaystyle i(u\ast_{b}v)$ $\displaystyle=i(b_{s_{1}}\dots
b_{s_{l-1}}\ast_{b}b_{r_{1}}\dots b_{r_{k}})p^{s_{l}}y+i(b_{s_{1}}\dots
b_{s_{l}}\ast_{b}b_{r_{1}}\dots b_{r_{k-1}})p^{r_{k}}y$ $\displaystyle\hskip
9.95863pt+i(b_{s_{2}}\dots b_{s_{l}}\ast_{b}b_{r_{2}}\dots
b_{r_{k}})p^{s_{l}+r_{k}}y.$

On the other hand, applying the definition of $\tau$ and $i$ gives
$\displaystyle\tau(\tau\circ i(u)\shuffle_{b}\tau\circ
i(v))=\tau(py^{s_{l}}\dots py^{s_{1}}\shuffle_{b}py^{r_{k}}\dots py^{r_{1}})$
$\displaystyle=\tau\Big{(}py^{s_{l}}(py^{s_{l-1}}\dots
py^{s_{1}}\shuffle_{b}py^{r_{k}}\dots py^{r_{1}})+py^{r_{k}}(py^{s_{l}}\dots
py^{s_{1}}\shuffle_{b}py^{r_{k-1}}\dots py^{r_{1}})$ $\displaystyle\hskip
9.95863pt+py^{s_{l}+r_{k}}(py^{s_{l-1}}\dots
py^{s_{1}}\shuffle_{b}py^{r_{k-1}}\dots py^{r_{1}})\Big{)}$
$\displaystyle=\tau\Big{(}\tau\circ i(b_{s_{1}}\dots
b_{s_{l-1}})\shuffle_{b}\tau\circ i(b_{r_{1}}\dots
b_{r_{k}})\Big{)}p^{s_{l}}y+\tau\Big{(}\tau\circ i(b_{s_{1}}\dots
b_{s_{l}})\shuffle_{b}\tau\circ i(b_{r_{1}}\dots
b_{r_{k-1}})\Big{)}p^{r_{k}}y$ $\displaystyle\hskip
9.95863pt+\tau\Big{(}\tau\circ i(b_{s_{1}}\dots
b_{s_{l-1}})\shuffle_{b}\tau\circ i(b_{r_{1}}\dots
b_{r_{k-1}})\Big{)}p^{s_{l}+r_{k}}y.$
So induction on the depth implies the claim. Next, assume that $s_{l}=0$ and
$r_{k}\geq 0$. Then we obtain

$\displaystyle i(u\ast_{b}v)$ $\displaystyle=i(b_{s_{1}}\dots
b_{s_{l-1}}\ast_{b}b_{r_{1}}\dots b_{r_{k}})y+i(b_{s_{1}}\dots
b_{s_{l}}\ast_{b}b_{r_{1}}\dots b_{r_{k-1}})p^{r_{k}}y,$

and on the other hand, using again the definition of $\tau$ and $i$

$\displaystyle\tau(\tau\circ i(u)\shuffle_{b}\tau\circ
i(v))=\tau(py^{s_{l}}\dots py^{s_{1}}\shuffle_{b}py^{r_{k}}\dots py^{r_{1}})$
$\displaystyle=\tau\Big{(}p(py^{s_{l-1}}\dots
py^{s_{1}}\shuffle_{b}py^{r_{k}}\dots py^{r_{1}})+py^{r_{k}}(py^{s_{l}}\dots
py^{s_{1}}\shuffle_{b}py^{r_{k-1}}\dots py^{r_{1}})\Big{)}$
$\displaystyle=\tau\Big{(}\tau\circ i(b_{s_{1}}\dots
b_{s_{l-1}})\shuffle_{b}\tau\circ i(b_{r_{1}}\dots
b_{r_{k}})\Big{)}y+\tau\Big{(}\tau\circ i(b_{s_{1}}\dots
b_{s_{l}})\shuffle_{b}\tau\circ i(b_{r_{1}}\dots
b_{r_{k-1}})\Big{)}p^{r_{k}}y.$

Again, induction on the depth implies the claim. ∎

By Proposition 11.3, there is injective algebra morphism

$\displaystyle\tau\circ i:(\mathbb{Q}\langle\mathcal{B}\rangle,\ast_{b})$
$\displaystyle\hookrightarrow(\mathbb{Q}\langle p,y\rangle,\shuffle_{b}),$
$\displaystyle b_{s_{1}}\dots b_{s_{l}}$ $\displaystyle\mapsto py^{s_{l}}\dots
py^{s_{1}}.$

In particular, the restriction of $\shuffle_{b}$ to
$\operatorname{im}(\tau\circ i)=\mathbb{Q}\mathbf{1}+p\mathbb{Q}\langle
p,y\rangle$ can be seen as a quasi-shuffle product. Denote

$\mathbb{Q}\langle p,y\rangle^{0}=\mathbb{Q}\mathbf{1}+p\mathbb{Q}\langle
p,y\rangle y.$

######  Theorem 11.4.

There is a $\tau$-invariant, surjective algebra morphism

$\displaystyle(\mathbb{Q}\langle p,y\rangle^{0},\shuffle_{b})$
$\displaystyle\to\mathcal{Z}_{q},$ $\displaystyle p^{s_{1}}y\dots p^{s_{l}}y$
$\displaystyle\mapsto\zeta_{q}(s_{1},\ldots,s_{l}).$

###### Proof.

Observe that by definition

$\displaystyle\zeta_{q}(i(u))=\zeta_{q}(u)\quad\text{ for all
}u\in\mathbb{Q}\langle\mathcal{B}\rangle^{0}.$ (11.1)

So as the balanced multiple q-zeta values form a spanning set of
$\mathcal{Z}_{q}$, we obtain surjectivity. Since $\tau(i(u))=i(\tau(u))$ for
each $u\in\mathbb{Q}\langle\mathcal{B}\rangle^{0}$, we deduce the
$\tau$-invariance from Theorem 10.4. Finally, we prove that the map is an
algebra morphism for $\shuffle_{b}$. For $s_{1},r_{1}\geq 1,\
s_{2},\ldots,s_{l},r_{2},\ldots,r_{k}\geq 0$, we obtain by applying the
$\tau$-invariance and (11.1)
$\displaystyle\zeta_{q}(py^{s_{l}}\dots py^{s_{1}})\zeta_{q}(py^{r_{k}}\dots
py^{r_{1}})$ $\displaystyle=\zeta_{q}(p^{s_{1}}y\dots
p^{s_{l}}y)\zeta_{q}(p^{r_{1}}y\dots p^{r_{k}}y)=\zeta_{q}(b_{s_{1}}\dots
b_{s_{l}})\zeta_{q}(b_{r_{1}}\dots b_{r_{k}}).$
Since $\zeta_{q}$ is an algebra morphism for the balanced quasi-shuffle
product $\ast_{b}$, we deduce

$\displaystyle\zeta_{q}(py^{s_{l}}\dots py^{s_{1}})\zeta_{q}(py^{r_{k}}\dots
py^{r_{1}})$ $\displaystyle=\zeta_{q}(b_{s_{1}}\dots
b_{s_{l}}\ast_{b}b_{r_{1}}\dots b_{r_{k}}).$

Applying Proposition 11.3 and then again the $\tau$-invariance gives

$\displaystyle\zeta_{q}(py^{s_{l}}\dots py^{s_{1}})\zeta_{q}(py^{r_{k}}\dots
py^{r_{1}})$ $\displaystyle=\zeta_{q}\big{(}\tau\big{(}\tau\circ
i(b_{s_{1}}\dots b_{s_{l}})\shuffle_{b}\tau\circ i(b_{r_{1}}\dots
b_{r_{k}})\big{)}\big{)}$ $\displaystyle=\zeta_{q}\big{(}\tau\circ
i(b_{s_{1}}\dots b_{s_{l}})\shuffle_{b}\tau\circ i(b_{r_{1}}\dots
b_{r_{k}})\big{)}.$

Finally, applying the definition of $\tau$ and $i$ yields the desired formula

$\displaystyle\zeta_{q}(py^{s_{l}}\dots py^{s_{1}})\zeta_{q}(py^{r_{k}}\dots
py^{r_{1}})$ $\displaystyle=\zeta_{q}\big{(}py^{s_{l}}\dots
py^{s_{1}}\shuffle_{b}py^{r_{k}}\dots py^{r_{1}}\big{)}.$

∎

Next, we will compute the limit of the balanced multiple q-zeta values for
$q\to 1$ for some special multi indices.

######  Proposition 11.5.

For a word $w=b_{\varepsilon_{1}}\dots b_{\varepsilon_{n}}b_{k_{1}}\dots
b_{k_{d}}$ in $\mathbb{Q}\langle\mathcal{B}\rangle^{0}$, where
$\varepsilon_{1},\ldots,\varepsilon_{n}\in\\{0,1\\}$, $\varepsilon_{n}=0$ and
$k_{1},\ldots,k_{d}\in\mathbb{Z}_{\geq 1}$, $k_{1}\geq 2$, we have

$\lim_{q\to
1}(1-q)^{\operatorname{wt}(w)}\zeta_{q}(w)=\zeta(x_{\varepsilon_{n}}\dots
x_{\varepsilon_{1}})\zeta(y_{k_{1}}\dots y_{k_{d}}).$

Here we set as explained in (1.2), (1.4),

$\displaystyle\zeta(x_{0}^{k_{1}-1}x_{1}\dots
x_{0}^{k_{d}-1}x_{1})=\zeta(k_{1},\ldots,k_{d}),\qquad\qquad\zeta(y_{k_{1}}\dots
y_{k_{d}})=\zeta(k_{1},\ldots,k_{d}),$

for all $k_{1},\ldots,k_{d}\in\mathbb{Z}_{\geq 1}$, $k_{1}\geq 2$.

###### Proof.

Let $\mathfrak{z}=(\mathfrak{z}_{d})_{d\geq 0}$ be the mould of the multiple
zeta values, so for each $d\geq 1$ we have

$\mathfrak{z}(X_{1},\ldots,X_{d})=\sum_{k_{1}\geq 2,k_{2},\ldots,k_{d}\geq
1}\zeta(k_{1},\ldots,k_{d})X_{1}^{k_{1}-1}\dots X_{d}^{k_{d}-1}.$

Similar to [BI22, Definition 1.3] define the conjugated multiple zeta values
$\xi(m_{1},\ldots,m_{d})$, $m_{1},\ldots,m_{d-1}\geq 0,\ m_{d}\geq 1$, by

$\displaystyle\mathfrak{z}(Y_{1}+\dots+Y_{d},\ldots,Y_{1}+Y_{2},Y_{1})=\sum_{m_{1},\ldots,m_{d-1}\geq
0,m_{d}\geq
1}\xi(m_{1},\ldots,m_{d})\frac{Y_{1}^{m_{1}}}{m_{1}!}\dots\frac{Y_{d}^{m_{d}}}{m_{d}!}.$
(11.2)

By [BI22, Theorem 4.18] and [BB22, Remark 6.18], the combinatorial bi-multiple
Eisenstein series satisfy

$\displaystyle\lim_{q\to
1}(1-q)^{m_{1}+\dots+m_{j}+j+k_{j+1}+\dots+k_{d}}G\binom{\hskip
8.5359pt1,\ldots,1,\hskip
5.69046ptk_{j+1},\ldots,k_{d}}{m_{1},\ldots,m_{j},\hskip
5.69046pt0,\ldots,0}=\xi(m_{1},\ldots,m_{j})\zeta(k_{j+1},\ldots,k_{d})$

for all $1\leq j\leq d$. We deduce from the definition of the balanced
multiple q-zeta values (Definition 10.1) that we have for
$\varepsilon_{1},\ldots,\varepsilon_{n}\in\\{0,1\\}$, $\varepsilon_{n}=0$ and
$k_{1},\ldots,k_{d}\in\mathbb{Z}_{\geq 1}$, $k_{1}\geq 2$

$\displaystyle\lim_{q\to
1}(1-q)^{n+k_{1}+\dots+k_{d}}\zeta_{q}(b_{\varepsilon_{1}}\dots
b_{\varepsilon_{n}}b_{k_{1}}\dots b_{k_{d}})$ $\displaystyle=\lim_{q\to
1}(1-q)^{n+k_{1}+\dots+k_{d}}G(\varphi_{\\#}^{-1}(b_{\varepsilon_{1}}\dots
b_{\varepsilon_{n}}b_{k_{1}}\dots b_{k_{d}}))$
$\displaystyle=\xi(\varphi_{\\#}^{-1}(b_{\varepsilon_{1}}\dots
b_{\varepsilon_{n}}))\zeta(k_{1},\ldots,k_{d}).$ (11.3)

Here $\xi(\varphi_{\\#}^{-1}(b_{\varepsilon_{1}}\dots b_{\varepsilon_{n}}))$
means that we first have to apply $\varphi_{\\#}$ to $b_{\varepsilon_{1}}\dots
b_{\varepsilon_{n}}$, which gives an element in $\mathbb{Q}\langle y_{1,m}\mid
m\geq 0\rangle$, and then we have to identify $\xi(y_{1,m_{1}}\dots
y_{1,m_{d}})=\xi(m_{1},\ldots,m_{d})$. On the level of generating series, this
means we have to apply $\\#_{Y}^{-1}$ (Definition 7.2), therefore we
substitute $Y_{1}\mapsto Y_{1},\ Y_{2}\mapsto Y_{2}-Y_{1},\ldots,Y_{d}\mapsto
Y_{d}-Y_{d-1}$ in the left-hand side of (11.2) and obtain

$\displaystyle\mathfrak{z}(Y_{d},Y_{d-1},\ldots,Y_{1})$
$\displaystyle=\sum_{m_{1},\ldots,m_{d-1}\geq 1,m_{d}\geq
2}\zeta(m_{d},\ldots,m_{1})Y_{1}^{m_{1}-1}\dots Y_{d}^{m_{d}-1}.$ (11.4)

Combining (11) and (11.4) gives the desired formula. ∎

######  Remark 11.6.

Similar to [BI22, Definition 4.17], one could define a regularized limit for
the balanced multiple q-zeta values by a certain regularization process in the
algebra $\mathbb{Q}\langle\mathcal{B}\rangle$ and obtains for each word
$w\in\mathbb{Q}\langle\mathcal{B}\rangle$

$\lim_{q\to
1}\textbf{}\\!^{*}(1-q)^{\operatorname{wt}(w)}\zeta_{q}(w)=\sum_{\begin{subarray}{c}uv=w\\\
u=b_{\varepsilon_{1}}\dots b_{\varepsilon_{n}},\
\varepsilon_{i}\in\\{0,1\\}\\\ v=b_{k_{1}}\dots b_{k_{d}},\
k_{i}\in\mathbb{N}\end{subarray}}\zeta^{\shuffle}(x_{\varepsilon_{n}}\dots
x_{\varepsilon_{1}})\zeta^{\ast}(y_{k_{1}}\dots y_{k_{d}}).$

Here $\zeta^{\shuffle}$ and $\zeta^{\ast}$ denote the shuffle and stuffle
regularized multiple zeta value map as given in (1.2), (1.4). In particular,
the regularized limit of the balanced multiple q-zeta value $\zeta_{q}(w)$
vanishes whenever the word $w$ cannot be decomposed as $w=uv$ with
$u\in\mathbb{Q}\langle b_{0},b_{1}\rangle$ and $v\in\mathbb{Q}\langle
b_{i}\mid i\geq 1\rangle$.

Finally, we want to describe the derivation
$q\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}q}$ on $\mathcal{Z}_{q}$
in terms of the balanced multiple q-zeta values. The obtained formula can be
seen as a weight-graded version of the derivation formula for the Schlesinger-
Zudilin multiple q-zeta values given in [Si15, Theorem 4.1].

######  Proposition 11.7.

We have for all $s_{1}\geq 1,s_{2},\ldots,s_{l}\geq 0$ that

$\displaystyle
q\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}q}\zeta_{q}(s_{1},\ldots,s_{l})=\sum_{i=1}^{l}\sum_{j=i}^{l}s_{i}\zeta_{q}(s_{1},\ldots,s_{i}+1,\ldots,s_{j},0,s_{j+1},\ldots,s_{l}).$

Observe that the derivation
$q\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}q}$ is a homogeneous
operator increasing the weight by $2$.

###### Proof.

By [BB22, Proposition 6.29] the generating series of the combinatorial bi-
multiple Eisenstein series satisfy

$\displaystyle
q\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}q}\mathfrak{G}_{d}\binom{X_{1},\ldots,X_{d}}{Y_{1},\ldots,Y_{d}}=\sum_{i=1}^{d}\frac{\partial}{\partial
X_{i}}\frac{\partial}{\partial
Y_{i}}\mathfrak{G}_{d}\binom{X_{1},\ldots,X_{d}}{Y_{1},\ldots,Y_{d}},\qquad
d\geq 1.$

By definition of the balanced multiple q-zeta values (Definition 10.1), we
obtain that
$\displaystyle\sum_{\begin{subarray}{c}k_{1},\dots,k_{d}\geq 1\\\
m_{1},\dots,m_{d}\geq
0\end{subarray}}q\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}q}\zeta_{q}(k_{1},\\{0\\}^{m_{1}},\ldots,k_{d},\\{0\\}^{m_{d}})X_{1}^{k_{1}-1}Y_{1}^{m_{1}}X_{2}^{k_{2}-1}(Y_{1}+Y_{2})^{m_{2}}\dots
X_{d}^{k_{d}-1}(Y_{1}+\dots+Y_{d})^{m_{d}}$ (11.5)
must be equal to
$\displaystyle\sum_{\begin{subarray}{c}k_{1},\dots,k_{d}\geq 1\\\
m_{1},\dots,m_{d}\geq
0\end{subarray}}\sum_{i=1}^{d}\zeta_{q}(k_{1},\\{0\\}^{m_{1}},\ldots,k_{d},\\{0\\}^{m_{d}})\frac{\partial}{\partial
X_{i}}\frac{\partial}{\partial
Y_{i}}X_{1}^{k_{1}-1}Y_{1}^{m_{1}}X_{2}^{k_{2}-1}(Y_{1}+Y_{2})^{m_{2}}$ (11.6)
$\displaystyle\hskip 341.43306pt\dots
X_{d}^{k_{d}-1}(Y_{1}+\dots+Y_{d})^{m_{d}}$
$\displaystyle=\sum_{\begin{subarray}{c}k_{1},\dots,k_{d}\geq 1\\\
m_{1},\dots,m_{d}\geq
0\end{subarray}}\sum_{i=1}^{d}\sum_{j=i}^{d}(k_{i}-1)m_{j}\zeta_{q}(k_{1},\\{0\\}^{m_{1}},\ldots,k_{d},\\{0\\}^{m_{d}})X_{1}^{k_{1}-1}Y_{1}^{m_{1}}X_{2}^{k_{2}-1}(Y_{1}+Y_{2})^{m_{2}}$
$\displaystyle\hskip 65.44142pt\dots
X_{i}^{k_{i}-2}(Y_{1}+\dots+Y_{i})^{m_{i}}\dots
X_{j}^{k_{j}-1}(Y_{1}+\dots+Y_{j})^{m_{j}-1}\dots
X_{d}^{k_{d}-1}(Y_{1}+\dots+Y_{d})^{m_{d}}.$
Coefficient comparison at
$X_{1}^{k_{1}-1}Y_{1}^{m_{1}}X_{2}^{k_{2}-1}(Y_{1}+Y_{2})^{m_{2}}\dots
X_{d}^{k_{d}-1}(Y_{1}+\dots+Y_{d})^{m_{d}}$ in (11.5) and (11.6) yields

$\displaystyle
q\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}q}\zeta_{q}(k_{1},\\{0\\}^{m_{1}},\ldots,k_{d},\\{0\\}^{m_{d}})$
$\displaystyle\hskip
56.9055pt=\sum_{i=1}^{d}\sum_{j=i}^{d}k_{i}(m_{j}+1)\zeta_{q}(k_{1},\\{0\\}^{m_{1}},\ldots,k_{i}+1,\\{0\\}^{m_{i}},\ldots,k_{j},\\{0\\}^{m_{j}+1},\ldots,k_{d},\\{0\\}^{m_{d}}),$

which is equivalent to the claimed formula. ∎

### Outlook.

In forthcoming articles we will formalize the balanced multiple q-zeta values
and obtain an algebra generated by symbols satisfying the balanced quasi-
shuffle product and $\tau$-invariance. Conjecturally, this algebra is equipped
with a Hopf algebra structure where the coproduct is given by a generalization
of Goncharov’s coproduct. A first step towards this is a Lie algebra structure
obtained in [Bu23]. Moreover, this algebra of symbols should give a complete
description of all relations in the algebra $\mathcal{Z}_{q}$ and by Theorem
7.10 also a description of all relations between multiple Eisenstein series.
By applying Racinet’s ideas for formal multiple zeta values ([Rac00]) to this
new algebra, we will obtain a formal version of the limit computation given in
Proposition 11.5. By [BI22] this map could be also seen as a formal version of
taking the constant term of q-series.

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