# q-analogues of multiple zeta values
and the formal double Eisenstein space

Henrik Bachmann  Graduate School of Mathematics, Nagoya University, Nagoya,
Japan<EMAIL_ADDRESS>
###### Abstract.

In this survey article, we discuss the algebraic structure of $q$-analogues of
multiple zeta values, which are closely related to derivatives of Eisenstein
series. Moreover, we introduce the formal double Eisenstein space, which
generalizes the formal double zeta space of Gangl, Kaneko, and Zagier. Using
the algebraic structure of $q$-analogues of multiple zeta values, we will
present a realization of this space. As an application, we will obtain purely
combinatorial proofs of identities among (quasi-)modular forms.

###### 2010 Mathematics Subject Classification:

Primary 11M32; Secondary 11F11

##  1\. Introduction

In this note, we discuss the algebraic structure of certain $q$-analogues of
multiple zeta values and their connection to modular forms. The purpose of
this survey article is to give a self-contained summary of some of the results
obtained in [B1] and [B2] and to motivate the works in progress [BB], [BKM]
and [BIM]. Some parts of those results were also announced in [BBK]. In
particular, we give an explanation of the formal double Eisenstein space,
which will be introduced and discussed in detail in [BKM] and which can be
seen as a generalization of the formal double zeta spaces introduced in [GKZ].
We will see that understanding the algebraic structure of $q$-analogues of
multiple zeta values allows us to write down an explicit realization of this
space. As an application of this, we will give some purely combinatorial
proofs of identities among (quasi-)modular forms such as the Ramanujan
differential equations.

Multiple zeta values are defined for $k_{1}\geq 2,k_{2},\dots,k_{r}\geq 1$ by

(1.1)
$\displaystyle\zeta(k_{1},\ldots,k_{r})=\sum_{m_{1}>\cdots>m_{r}>0}\frac{1}{m_{1}^{k_{1}}\cdots
m_{r}^{k_{r}}}\,.$

These values are known to satisfy a collection of relations called the double
shuffle relations. For example, in smallest depth and $k_{1},k_{2}\geq 2$
these are given by

(1.2)
$\displaystyle\begin{split}\zeta(k_{1})\zeta(k_{2})&=\zeta(k_{1},k_{2})+\zeta(k_{2},k_{1})+\zeta(k_{1}+k_{2})\\\
&=\sum_{j=2}^{k_{1}+k_{2}-1}\left(\binom{j-1}{k_{1}-1}+\binom{j-1}{k_{2}-1}\right)\zeta(j,k_{1}+k_{2}-j)\,.\end{split}$

The first one, called the stuffle product, is a direct consequence of the
definition (1.1). The second one, called the shuffle product, can be proven
either by using an iterated integral expression for multiple zeta values or by
using partial fraction decomposition. Motivated by the relations (1.2) Gangl,
Kaneko and Zagier defined in [GKZ] the _formal double zeta space_ for $k\geq
1$ by the $\mathbb{Q}$-vector space

$\displaystyle\mathcal{D}_{k}=\mathchoice{\text{\raise
4.30554pt\hbox{$\big{\langle}Z_{k},Z_{k_{1},k_{2}},P_{k_{1},k_{2}}\mid
k_{1}+k_{2}=k,k_{1},k_{2}\geq 1\big{\rangle}_{\mathbb{Q}}$}\Big{/}\lower
4.30554pt\hbox{$\eqref{eq:dzrel}$}}}{\big{\langle}Z_{k},Z_{k_{1},k_{2}},P_{k_{1},k_{2}}\mid
k_{1}+k_{2}=k,k_{1},k_{2}\geq
1\big{\rangle}_{\mathbb{Q}}\,/\,\eqref{eq:dzrel}}{\big{\langle}Z_{k},Z_{k_{1},k_{2}},P_{k_{1},k_{2}}\mid
k_{1}+k_{2}=k,k_{1},k_{2}\geq
1\big{\rangle}_{\mathbb{Q}}\,/\,\eqref{eq:dzrel}}{\big{\langle}Z_{k},Z_{k_{1},k_{2}},P_{k_{1},k_{2}}\mid
k_{1}+k_{2}=k,k_{1},k_{2}\geq
1\big{\rangle}_{\mathbb{Q}}\,/\,\eqref{eq:dzrel}}\,,$

which is spanned by formal symbols $Z_{k},Z_{k_{1},k_{2}},P_{k_{1},k_{2}}$
satisfying the relations

(1.3)
$\displaystyle\begin{split}P_{k_{1},k_{2}}&=Z_{k_{1},k_{2}}+Z_{k_{2},k_{1}}+Z_{k_{1}+k_{2}}\\\
&=\sum_{j=1}^{k_{1}+k_{2}-1}\left(\binom{j-1}{k_{1}-1}+\binom{j-1}{k_{2}-1}\right)Z_{j,k_{1}+k_{2}-j}\,.\end{split}$

In [GKZ] different realizations of this space are given, where for a
$\mathbb{Q}$-vector space $A$ a realization of $\mathcal{D}_{k}$ in $A$ is
defined as an element in $\operatorname{Hom}_{\mathbb{Q}}(\mathcal{D}_{k},A)$.
By (1.2) together with some regularization one can show that for
$A=\mathbb{R}$ we obtain a realization of $\mathcal{D}_{k}$ by

(1.4) $\displaystyle\begin{split}Z_{k}&\longmapsto\begin{cases}\zeta(k)&k\geq
3\\\ 0&k=1,2\end{cases}\,,\\\
Z_{k_{1},k_{2}}&\longmapsto\begin{cases}\zeta(k_{1},k_{2})&k_{1}\geq 2\\\
-\zeta(k_{2},1)-\zeta(k_{2}+1)&k_{1}=1,k_{2}\geq 2\\\
0&k_{1}=k_{2}=1\end{cases}\,,\\\
P_{k_{1},k_{2}}&\longmapsto\begin{cases}\zeta(k_{1})\zeta(k_{2})&k_{1},k_{2}\neq
1\\\ 0&k_{1}=1\vee k_{2}=1\end{cases}\,.\end{split}$

The Riemann zeta values $\zeta(k)$ also appear as the constant term of the
Eisenstein series defined for $k\geq 2$ by

(1.5) $\displaystyle G_{k}(\tau)=\zeta(k)+\frac{(-2\pi
i)^{k}}{(k-1)!}\sigma_{k-1}(n)q^{n}\,,$

where $\tau\in\mathbb{H}=\\{z\in\mathbb{C}\mid\operatorname{Im}(z)>0\\}$,
$q=e^{2\pi i\tau}$ and $\sigma_{k-1}(n)=\sum_{d\mid n}d^{k-1}$. For even
$k\geq 4$ these are the first non-trivial examples of modular forms of weight
$k$ for the full modular group. In [GKZ] it was observed, that the relations
(1.2) can be lifted to the space of holomorphic functions, such that the
Riemann zeta values gets replaced by the Eisenstein series and the double zeta
values by so-called double Eisenstein series $G_{k_{1},k_{r}}$. In other words
they showed that for $k\geq 4$ there exists a realization of $\mathcal{D}_{k}$
in the space of holomorphic functions in the upper half plane with
$Z_{k}\mapsto G_{k}$ (where $G_{1}$ is given by (1.5) without the constant
term), $Z_{k_{1},k_{2}}\mapsto G_{k_{1},k_{r}}$ and

(1.6) $\displaystyle P_{k_{1},k_{2}}$ $\displaystyle\longmapsto
G_{k_{1}}G_{k_{2}}+\frac{\delta_{k_{1},2}}{2k_{2}}G^{\prime}_{k_{2}}+\frac{\delta_{k_{2},2}}{2k_{1}}G^{\prime}_{k_{1}}\,.$

Taking the constant term of this realization gives the multiple zeta value
realization (1.4). But in contrast to (1.4) the image of $P_{k_{1},k_{2}}$ in
(1.6) is not given exactly by the product of the depth one objects, since also
derivatives of Eisenstein series appear. This has to do with the fact that
$G_{2}$ is not modular, and it is one of the motivations for the formal double
Eisenstein space, which will be introduced in [BKM] and which we will explain
at the end of this note. In this space, we will also consider ”formal
derivatives” of our objects. As it will turn out, this gives an analogue of
the formal double zeta space, which is better suited for Eisenstein series in
the sense that we will find a realization of this space where the formal
symbol of the product gets mapped exactly to the product of Eisenstein series.
It can also be seen as a generalization of the work of Gangl, Kaneko, Zagier
from modular forms to quasi-modular forms.

### Acknowledgment

The author would like to thank the organizers of the Waseda number theory
Conference 2021 for giving him the opportunity to present parts of the results
of [BB], [BIM] and [BKM] during the conference. These projects are partially
supported by JSPS KAKENHI Grant 19K14499.

##  2\. $q$-analogues of multiple zeta values

This section recalls the $q$-analogues of multiple zeta values, which were
introduced in [B1]. Parts of the results are contained in [BK] and detailed
information can be found in [B2]. Most of the time we will only consider the
depth two case, but everything done in this section also has a generalization
to arbitrary depths.

######  Definition 2.1.

A _modified q-analogue of weight $k$_ of a complex number $c\in\mathbb{C}$ is
a q-series $f\in\mathbb{C}[[q]]$, such that

$\displaystyle\lim_{q\rightarrow 1}(1-q)^{k}f=c\,.$

Here and in the following by $q\rightarrow 1$, we always mean the limit of $q$
on the real axis with $0<q<1$. Usually, our objects are just formal $q$-series
in $\mathbb{Q}[[q]]$, but all appearing $q$-series in this note can also be
seen as holomorphic functions in the unit disc ($|q|<1$) or as holomorphic
functions in the upper half-plane by setting $q=e^{2\pi i\tau}$. We will not
distinguish these three points of view in the following.

######  Proposition 2.2.

Let $f(\tau)=a_{0}+\sum_{n\geq 1}a_{n}q^{n}$ be a modular form of weight
$k$.Then $f$ is a modified q-analogue of $(2\pi i)^{k}a_{0}$.

###### Proof.

If $f$ is modular of weight $k$ then $f(-\frac{1}{\tau})=\tau^{k}f(\tau)$,
i.e.

$\displaystyle\lim_{q\rightarrow 1}(1-q)^{k}f(q)$
$\displaystyle=\lim_{\tau\rightarrow 0}((2\pi
i\tau)^{k}+O(\tau^{k+1}))f(\tau)=\lim_{\tau\rightarrow 0}(2\pi
i)^{k}f\left(-\frac{1}{\tau}\right)$ $\displaystyle=\lim_{\tau\rightarrow
i\infty}(2\pi i)^{k}f(\tau)=(2\pi i)^{k}a_{0}\,.$

∎

In particular the normalized Eisenstein series

(2.1) $\displaystyle\widetilde{G}_{k}(\tau):=\frac{1}{(2\pi
i)^{k}}G_{k}(\tau)=-\frac{B_{k}}{2k!}+\frac{1}{(k-1)!}\sum_{n>0}\sigma_{k-1}(n)q^{n}$

are modified $q$-analogue (of weight $k$) of $\zeta(k)$ for even $k\geq 4$.
But we will see that we have for any $k\geq 2$

$\displaystyle\lim_{q\rightarrow
1}(1-q)^{k}\underbrace{\frac{1}{(k-1)!}\sum_{n>0}\sigma_{k-1}(n)q^{n}}_{g(k):=}=\zeta(k)\,.$

In the following we will consider a multiple version of the $q$-series
$g(k)=\sum_{\begin{subarray}{c}m>0\\\
n>0\end{subarray}}\frac{n^{k-1}}{(k-1)!}q^{mn}$.

######  Definition 2.3.

For $k_{1},\dots k_{r}\geq 1$ we define the $q$-series
$g(k_{1},\dots,k_{r})\in\mathbb{Q}[[q]]$ by

$\displaystyle
g(k_{1},\dots,k_{r})=\sum_{\begin{subarray}{c}m_{1}>\dots>m_{r}>0\\\
n_{1},\dots,n_{r}>0\end{subarray}}\frac{n_{1}^{k_{1}-1}}{(k_{1}-1)!}\dots\frac{n_{r}^{k_{r}-1}}{(k_{r}-1)!}q^{m_{1}n_{1}+\dots+m_{r}n_{r}}\,.$

Notice that for $r=1$ these are, up to the constant term, exactly the
Eisenstein series $\widetilde{G}_{k}$. Moreover, they are modified q-analogues
of multiple zeta values.

######  Proposition 2.4.

For $k_{1}\geq 2$, $k_{2},\dots,k_{r}\geq 1$ the $g(k_{1},\dots,k_{r})$ are
modified q-analogues of $\zeta(k_{1},\dots,k_{r})$, i.e.

$\displaystyle\lim\limits_{q\rightarrow
1}(1-q)^{k_{1}+\dots+k_{r}}g(k_{1},\dots,k_{r})=\zeta(k_{1},\dots,k_{r})\,.$

###### Proof.

First rewrite

$\displaystyle
g(k_{1},\dots,k_{r})=\sum_{m_{1}>\dots>m_{r}>0}\frac{P_{k_{1}}(q^{m_{1}})}{(1-q^{m_{1}})^{k_{1}}}\dots\frac{P_{k_{r}}(q^{m_{r}})}{(1-q^{m_{r}})^{k_{r}}}\,,$

where the polynomials $P_{k}$ are defined by
$\frac{P_{k}(X)}{(1-X)^{k}}:=\sum_{n>0}\frac{n^{k-1}}{(k-1)!}X^{k-1}$. These
are, up to a normalization, the _Eulerian polynomials_ and they satisfy
$P_{k}(1)=1$ from which we get after using some elementary convergence
criteria that

$\displaystyle\lim\limits_{q\rightarrow
1}(1-q)^{k_{1}+\dots+k_{r}}g(k_{1},\dots,k_{r})=\lim\limits_{q\rightarrow
1}\sum_{m_{1}>\dots>m_{r}>0}\prod_{j=1}^{r}\left(\frac{1-q}{1-q^{m_{j}}}\right)^{k_{j}}P_{k_{j}}(q^{m_{j}})=\sum_{m_{1}>\dots>m_{r}>0}\prod_{j=1}^{r}\frac{1}{m_{j}^{k_{j}}}\,.$

∎

######  Proposition 2.5.

For $k_{1},k_{2}\geq 1$ we have

(2.2) $\displaystyle
g(k_{1})g(k_{2})=g(k_{1},k_{2})+g(k_{2},k_{1})+g(k_{1}+k_{2})+\sum_{j=1}^{k_{1}+k_{2}-1}\lambda^{j}_{k_{1},k_{2}}\,g(j)\,,$

where the rational numbers $\lambda^{j}_{k_{1},k_{2}}$ are given by

(2.3)
$\displaystyle\lambda^{j}_{k_{1},k_{2}}=\left((-1)^{k_{1}-1}\binom{k_{1}+k_{2}-1-j}{k_{2}-j}+(-1)^{k_{2}-1}\binom{k_{1}+k_{2}-1-j}{k_{1}-j}\right)\frac{B_{k_{1}+k_{2}-j}}{(k_{1}+k_{2}-j)!}\,.$

###### Proof.

The proof uses the generating function $L_{m}(X)=\sum_{k\geq
1}\frac{P_{k}(q^{m})}{(1-q^{m})^{k}}X^{k}=\frac{e^{X}q^{m}}{1-e^{X}q^{m}}$. By
direct calculation one then checks that

$\displaystyle
L_{m}(X)L_{m}(Y)=\frac{1}{e^{X-Y}-1}L_{m}(X)+\frac{1}{e^{Y-X}-1}L_{m}(Y),$

and uses $\sum_{n=0}^{\infty}B_{n}\frac{x^{n}}{n!}=\frac{x}{e^{x}-1}$ to show

$\displaystyle\frac{P_{k_{1}}(q^{m})}{(1-q^{m})^{k_{1}}}\frac{P_{k_{2}}(q^{m})}{(1-q^{m})^{k_{2}}}$
$\displaystyle=\frac{P_{k_{1}+k_{2}}(q^{m})}{(1-q^{m})^{k_{1}+k_{2}}}+\sum_{j=1}^{k_{1}+k_{2}-1}\lambda^{j}_{k_{1},k_{2}}\frac{P_{j}(q^{m})}{(1-q^{m})^{j}}\,.$

The statement then follows since

$\displaystyle g(k_{1})g(k_{2})$
$\displaystyle=\sum_{m_{1}>0}\frac{P_{k_{1}}(q^{m_{1}})}{(1-q^{m_{1}})^{k_{1}}}\sum_{m_{2}>0}\frac{P_{k_{2}}(q^{m_{2}})}{(1-q^{m_{2}})^{k_{2}}}$
$\displaystyle=\left(\sum_{m_{1}>m_{2}>0}+\sum_{m_{2}>m_{1}>0}+\sum_{m_{1}=m_{2}>0}\right)\frac{P_{k_{1}}(q^{m_{1}})}{(1-q^{m_{1}})^{k_{1}}}\frac{P_{k_{2}}(q^{m_{2}})}{(1-q^{m_{2}})^{k_{2}}}\,.$

∎

Except for the lower weight terms the product (2.2) looks exactly like the
stuffle product
$\zeta(k_{1})\zeta(k_{2})=\zeta(k_{1},k_{2})+\zeta(k_{2},k_{1})+\zeta(k_{1}+k_{2})$
of zeta values. Notice that this formula for zeta values follows from
Proposition 2.4 for $k_{1},k_{2}\geq 2$ by multiplying (2.2) with
$(1-q)^{k_{1}+k_{2}}$ and then taking the limit $q\rightarrow 1$ since the
lower weight terms vanish. So a natural question is if there is also an
analogue of the shuffle product for the $q$-series $g$. Before answering this
question we consider the operator $q\frac{d}{dq}$ (which corresponds to
$\frac{1}{2\pi i}\frac{d}{d\tau}$ for $q=e^{2\pi i\tau}$).

$\displaystyle
q\frac{d}{dq}g_{k}(q)=q\frac{d}{dq}\sum_{\begin{subarray}{c}m>0\\\
n>0\end{subarray}}\frac{n^{k-1}}{(k-1)!}q^{mn}=\sum_{\begin{subarray}{c}m>0\\\
n>0\end{subarray}}\frac{mn^{k}}{(k-1)!}q^{mn}\,.$

We see that after taking the derivative we also have a $m$ appearing in the
numerator. Moreover if we would take the $d$-th derivative we would get

$\displaystyle\left(q\frac{d}{dq}\right)^{d}g_{k}(q)=\sum_{\begin{subarray}{c}m>0\\\
n>0\end{subarray}}\frac{m^{d}n^{k+d-1}}{(k-1)!}q^{mn}\,.$

This leads us to define $g$ in a more general way.

######  Definition 2.6.

For $k_{1},\dots k_{r}\geq 1,d_{1},\dots,d_{r}\geq 0$ define the $q$-series

$\displaystyle
g\genfrac{(}{)}{0.0pt}{}{k_{1},\dots,k_{r}}{d_{1},\dots,d_{r}}=\sum_{\begin{subarray}{c}m_{1}>\dots>m_{r}>0\\\
n_{1},\dots,n_{r}>0\end{subarray}}\frac{m_{1}^{d_{1}}n_{1}^{k_{1}-1}}{(k_{1}-1)!}\dots\frac{m_{r}^{d_{r}}n_{r}^{k_{r}-1}}{(k_{r}-1)!}q^{m_{1}n_{1}+\dots+m_{r}n_{r}}\,.$

We say that this has _weight_ $k_{1}+\dots+k_{r}+d_{1}+\dots+d_{r}$ and
_depth_ $r$.

Clearly these generalize the $q$-series in Definition 2.3 since we have

$\displaystyle
g(k_{1},\dots,k_{r})=g\genfrac{(}{)}{0.0pt}{}{k_{1},\dots,k_{r}}{0,\dots,0}\,.$

Notice that again for $r=1$ these $q$-series are essentially the derivatives
of Eisenstein series since for $k>d$ we have

$\displaystyle
g\genfrac{(}{)}{0.0pt}{}{k}{d}=\frac{(k-d-1)!}{(k-1)!}\left(q\frac{d}{dq}\right)^{d}\widetilde{G}_{k-d}+\delta_{d,0}\frac{B_{k}}{2k!}\,.$

With the same idea as before it is easy to see that we have

$\displaystyle
q\frac{d}{dq}g\genfrac{(}{)}{0.0pt}{}{k_{1},\dots,k_{r}}{d_{1},\dots,d_{r}}=\sum_{j=1}^{r}k_{j}g\genfrac{(}{)}{0.0pt}{}{k_{1},\dots.k_{j}+1,\dots,k_{r}}{d_{1},\dots,d_{j}+1,\dots,d_{r}}\,.$

But the main reason for introducing this more general definition is not that
we can talk about derivatives, but rather we will see that this enables us to
define an analogue of the shuffle product. Before doing this we will see that
the formula for the stuffle product (Proposition 2.4) generalizes easily.

######  Proposition 2.7 (Stuffle product analogue).

For $k_{1},k_{2}\geq 1,d_{1},d_{2}\geq 0$ we have

(2.4) $\displaystyle
g\genfrac{(}{)}{0.0pt}{}{k_{1}}{d_{1}}g\genfrac{(}{)}{0.0pt}{}{k_{2}}{d_{2}}=$
$\displaystyle
g\genfrac{(}{)}{0.0pt}{}{k_{1},k_{2}}{d_{1},d_{2}}+g\genfrac{(}{)}{0.0pt}{}{k_{2},k_{1}}{d_{2},d_{1}}+g\genfrac{(}{)}{0.0pt}{}{k_{1}+k_{2}}{d_{1}+d_{2}}+\sum_{j=1}^{k_{1}+k_{2}-1}\lambda^{j}_{k_{1},k_{2}}\,g\genfrac{(}{)}{0.0pt}{}{j}{d_{1}+d_{2}}\,,$

where the $\lambda^{j}_{k_{1},k_{2}}$ are given by (2.3).

###### Proof.

Similar proof as Proposition 2.5 since the exponents of the $m_{j}$ just add
up.∎

Let $\operatorname{Part}_{r}(N)$ denote the set of partitions of $N$ made out
of $r$ different parts. For example, $11=4+4+1+1+1$ is a partition of $N=11$
made out of $r=2$ different parts $(m_{1},m_{2})=(4,1)$ with multiplicities
$(n_{1},n_{2})=(2,3)$, i.e. $N=m_{1}n_{1}+m_{2}n_{2}$. Any element
$\lambda\in\operatorname{Part}_{r}(N)$ can be represented by a Young diagram

$\lambda=$$m_{1}$$m_{2}$$m_{r-1}$$m_{r}$$n_{1}$$n_{2}$$n_{r-1}$$n_{r}$

where $N=m_{1}n_{1}+\dots+m_{r}n_{r}$ and $m_{1}>\dots>m_{r}>0$,
$n_{1},\dots,n_{r}>0$. In particular we can write the coefficient of $q^{N}$
in the $q$-series $g$ as a sum over all elements in
$\operatorname{Part}_{r}(N)$

$\displaystyle
g\genfrac{(}{)}{0.0pt}{}{k_{1},\dots,k_{r}}{d_{1},\dots,d_{r}}=\sum_{\begin{subarray}{c}m_{1}>\dots>m_{r}>0\\\
n_{1},\dots,n_{r}>0\end{subarray}}\underbrace{\frac{m_{1}^{d_{1}}n_{1}^{k_{1}-1}}{(k_{1}-1)!}\dots\frac{m_{r}^{d_{r}}n_{r}^{k_{r}-1}}{(k_{r}-1)!}}_{f(\lambda):=}q^{m_{1}n_{1}+\dots+m_{r}n_{r}}=\sum_{N>0}\left(\sum_{\lambda\in\operatorname{Part}_{r}(N)}f(\lambda)\right)q^{N}\,,$

where $f:\operatorname{Part}_{r}(N)\rightarrow\mathbb{Q}$ sends a partition
$\lambda$, given as above, to a polynomial in the $m_{j}$ and $n_{j}$
depending on the $k_{j}$ and $d_{j}$. On $\operatorname{Part}_{r}(N)$ we have
an involution given by the conjugation $\rho$ of Young diagrams, which
correspond to reflecting the Young diagram along the diagonal

$\lambda=$$m_{1}$$m_{2}$$m_{r-1}$$m_{r}$$n_{1}$$n_{2}$$n_{r-1}$$n_{r}$$\rho$$=\rho(\lambda)$$n_{1}+\dots+n_{r}$$n_{1}+\dots+n_{r-1}$$n_{1}+n_{2}$$n_{1}$$m_{r}$$m_{r-1}-m_{r}$$m_{2}-m_{3}$$m_{1}-m_{2}$

Since we sum over all elements in $\operatorname{Part}_{r}(N)$ we obtain

$\displaystyle
g\genfrac{(}{)}{0.0pt}{}{k_{1},\dots,k_{r}}{d_{1},\dots,d_{r}}=\sum_{N>0}\left(\sum_{\lambda\in\operatorname{Part}_{r}(N)}f(\lambda)\right)q^{N}=\sum_{N>0}\left(\sum_{\lambda\in\operatorname{Part}_{r}(N)}f(\rho(\lambda))\right)q^{N}\,.$

The $f(\rho(\lambda))$ is again a polynomial in
$m_{1},\dots,m_{r},n_{1},\dots,n_{r}$ and therefore the right-hand side of
above equation can be written again as a linear combination of the $q$-series
$g$ in the same depth and weight. We call the linear relation obtained from
this _partition relation_. These linear relations can be written done
explicitly in all depths, and we give the explicit formula for depth one and
two in the following.

######  Proposition 2.8 (Partition relation).

  1. i)

For $k\geq 1,d\geq 1$ we have

$\displaystyle
g\genfrac{(}{)}{0.0pt}{}{k}{d}=\frac{d!}{(k-1)!}g\genfrac{(}{)}{0.0pt}{}{d+1}{k-1}\,.$

  2. ii)

For $k_{1},k_{2}\geq 1,d_{1},d_{2}\geq 0$ we have

$\displaystyle
g\genfrac{(}{)}{0.0pt}{}{k_{1},k_{2}}{d_{1},d_{2}}=\sum_{\begin{subarray}{c}0\leq
a\leq d_{1}\\\ 0\leq b\leq
k_{2}-1\end{subarray}}\frac{(-1)^{b}}{a!\,b!}\frac{d_{1}!}{(k_{1}-1)!}\frac{(d_{2}+a)!}{(k_{2}-1-b)!}\,\,g\genfrac{(}{)}{0.0pt}{}{d_{2}+1+a,\,d_{1}+1-a}{k_{2}-1-b,\,k_{1}-1+b}\,.$

######  Example 2.9.

Combining the stuffle product (Proposition 2.7) and the partition relation
(Proposition 2.8) gives us another way of expressing the product of two
$q$-series $g$:

$\displaystyle g(2)g(3)$
$\displaystyle=g\genfrac{(}{)}{0.0pt}{}{2}{0}g\genfrac{(}{)}{0.0pt}{}{3}{0}\overset{\text{Prop.}\ref{prop:partrel}}{=}\frac{1}{2}g\genfrac{(}{)}{0.0pt}{}{1}{1}g\genfrac{(}{)}{0.0pt}{}{1}{2}$
$\displaystyle\overset{\text{Prop.}\ref{prop:stufflebig}}{=}\frac{1}{2}\left(g\genfrac{(}{)}{0.0pt}{}{1,1}{1,2}+g\genfrac{(}{)}{0.0pt}{}{1,1}{2,1}+g\genfrac{(}{)}{0.0pt}{}{2}{3}-g\genfrac{(}{)}{0.0pt}{}{1}{3}\right)$
$\displaystyle\overset{\text{Prop.}\ref{prop:partrel}}{=}g\genfrac{(}{)}{0.0pt}{}{2,3}{0,0}+3g\genfrac{(}{)}{0.0pt}{}{3,2}{0,0}+6g\genfrac{(}{)}{0.0pt}{}{4,1}{0,0}+3g\genfrac{(}{)}{0.0pt}{}{4}{1}-3g\genfrac{(}{)}{0.0pt}{}{4}{0}.$

Using
$q\frac{d}{dq}g\genfrac{(}{)}{0.0pt}{}{3}{0}=3g\genfrac{(}{)}{0.0pt}{}{4}{1}$
we obtain

$\displaystyle g(2)g(3)$
$\displaystyle=g(2,3)+3g(3,2)+6g(4,1)-3g(4)+q\frac{d}{dq}g(3)\,.$

This formula looks similar to the shuffle product of the corresponding zeta
values, which is given by

$\displaystyle\zeta(2)\zeta(3)=\zeta(2,3)+3\zeta(3,2)+6\zeta(4,1)\,.$

But now we do not just have an additional lower weight term
$3g\genfrac{(}{)}{0.0pt}{}{4}{0}$, but also the term
$3g\genfrac{(}{)}{0.0pt}{}{4}{1}$, which is of weight $5$. Though these terms
vanish after multiplying with $(1-q)^{5}$ and sending $q\rightarrow 1$, this
additional weight $5$ term is the reason why there is no realization of the
formal double zeta space $\mathcal{D}_{k}$ with $Z_{k}\mapsto G_{k}$ and where
the image of $P_{k_{1},k_{2}}$ is given exactly by products of Eisenstein
series.

######  Proposition 2.10 (Shuffle product analogue).

For $k_{1},k_{2}\geq 1,d_{1},d_{2}\geq 0$ we have

$\displaystyle
g\genfrac{(}{)}{0.0pt}{}{k_{1}}{d_{1}}g\genfrac{(}{)}{0.0pt}{}{k_{2}}{d_{2}}=$
$\displaystyle\sum_{\begin{subarray}{c}l_{1}+l_{2}=k_{1}+k_{2}\\\
e_{1}+e_{2}=d_{1}+d_{2}\end{subarray}}\left(\binom{l_{1}-1}{k_{1}-1}\binom{d_{1}}{e_{1}}(-1)^{d_{1}-e_{1}}+\binom{l_{1}-1}{k_{2}-1}\binom{d_{2}}{e_{1}}(-1)^{d_{2}-e_{1}}\right)g\genfrac{(}{)}{0.0pt}{}{l_{1},l_{2}}{e_{1},e_{2}}$
$\displaystyle+\frac{d_{1}!d_{2}!}{(d_{1}+d_{2}+1)!}\binom{k_{1}+k_{2}-2}{k_{1}-1}g\genfrac{(}{)}{0.0pt}{}{k_{1}+k_{2}-1}{d_{1}+d_{2}+1}$
$\displaystyle+d_{1}!d_{2}!\binom{k_{1}+k_{2}-2}{k_{1}-1}\sum_{j=1}^{d_{1}+d_{2}+1}\frac{\lambda^{j}_{d_{1}+1,d_{2}+1}}{(j-1)!}\,g\genfrac{(}{)}{0.0pt}{}{k_{1}+k_{2}-1}{j-1}\,.$

###### Proof.

Similar as in Example 2.9 we first use Proposition 2.8 i), then Proposition
2.7 to evaluate the product and then use again Proposition 2.8 i) and ii). ∎

##  3\. Formal double Eisenstein space

In this section, we will introduce the formal double Eisenstein space which
will be discussed in more detail in [BKM]. This space will be spanned by
formal symbols satisfying the same relations as the $q$-series $g$ except that
we just consider the homogeneous weight parts and ignore the lower weight
terms. The motivation behind this is that, for example, in depth one, the main
objects we are interested in are not the $g(k)$ but the Eisenstein series
$\widetilde{G}_{k}=-\frac{B_{k}}{2k!}+g(k)$ which do not satisfy algebraic
relations in mixed weight.

######  Definition 3.1.

We define for $K\geq 1$ the _formal double Eisenstein space_ of weight $K$ as

$\displaystyle\mathcal{E}_{K}=\mathchoice{\text{\raise
4.30554pt\hbox{$\Big{\langle}G\genfrac{(}{)}{0.0pt}{}{k}{d},G\genfrac{(}{)}{0.0pt}{}{k_{1},k_{2}}{d_{1},d_{2}},P\genfrac{(}{)}{0.0pt}{}{k_{1},k_{2}}{d_{1},d_{2}}\mid\begin{subarray}{c}k+d=k_{1}+k_{2}+d_{1}+d_{2}=K\\\
k,k_{1},k_{2}\geq 1,\,d,d_{1},d_{2}\geq
0\end{subarray}\Big{\rangle}_{\mathbb{Q}}$}\Big{/}\lower
4.30554pt\hbox{$\eqref{eq:derel}$}}}{\Big{\langle}G\genfrac{(}{)}{0.0pt}{}{k}{d},G\genfrac{(}{)}{0.0pt}{}{k_{1},k_{2}}{d_{1},d_{2}},P\genfrac{(}{)}{0.0pt}{}{k_{1},k_{2}}{d_{1},d_{2}}\mid\begin{subarray}{c}k+d=k_{1}+k_{2}+d_{1}+d_{2}=K\\\
k,k_{1},k_{2}\geq 1,\,d,d_{1},d_{2}\geq
0\end{subarray}\Big{\rangle}_{\mathbb{Q}}\,/\,\eqref{eq:derel}}{\Big{\langle}G\genfrac{(}{)}{0.0pt}{}{k}{d},G\genfrac{(}{)}{0.0pt}{}{k_{1},k_{2}}{d_{1},d_{2}},P\genfrac{(}{)}{0.0pt}{}{k_{1},k_{2}}{d_{1},d_{2}}\mid\begin{subarray}{c}k+d=k_{1}+k_{2}+d_{1}+d_{2}=K\\\
k,k_{1},k_{2}\geq 1,\,d,d_{1},d_{2}\geq
0\end{subarray}\Big{\rangle}_{\mathbb{Q}}\,/\,\eqref{eq:derel}}{\Big{\langle}G\genfrac{(}{)}{0.0pt}{}{k}{d},G\genfrac{(}{)}{0.0pt}{}{k_{1},k_{2}}{d_{1},d_{2}},P\genfrac{(}{)}{0.0pt}{}{k_{1},k_{2}}{d_{1},d_{2}}\mid\begin{subarray}{c}k+d=k_{1}+k_{2}+d_{1}+d_{2}=K\\\
k,k_{1},k_{2}\geq 1,\,d,d_{1},d_{2}\geq
0\end{subarray}\Big{\rangle}_{\mathbb{Q}}\,/\,\eqref{eq:derel}}\,,$

where we divide out the following relations

(3.1)
$\displaystyle\begin{split}P\genfrac{(}{)}{0.0pt}{}{k_{1},k_{2}}{d_{1},d_{2}}&=G\genfrac{(}{)}{0.0pt}{}{k_{1},k_{2}}{d_{1},d_{2}}+G\genfrac{(}{)}{0.0pt}{}{k_{2},k_{1}}{d_{2},d_{1}}+G\genfrac{(}{)}{0.0pt}{}{k_{1}+k_{2}}{d_{1}+d_{2}}\\\
&=\sum_{\begin{subarray}{c}l_{1}+l_{2}=k_{1}+k_{2}\\\
e_{1}+e_{2}=d_{1}+d_{2}\\\ l_{1},l_{2}\geq 1,e_{1},e_{2}\geq
0\end{subarray}}\left(\binom{l_{1}-1}{k_{1}-1}\binom{d_{1}}{e_{1}}(-1)^{d_{1}-e_{1}}+\binom{l_{1}-1}{k_{2}-1}\binom{d_{2}}{e_{1}}(-1)^{d_{2}-e_{1}}\right)G\genfrac{(}{)}{0.0pt}{}{l_{1},l_{2}}{e_{1},e_{2}}\\\
&+\frac{d_{1}!d_{2}!}{(d_{1}+d_{2}+1)!}\binom{k_{1}+k_{2}-2}{k_{1}-1}G\genfrac{(}{)}{0.0pt}{}{k_{1}+k_{2}-1}{d_{1}+d_{2}+1}\,.\end{split}$

The reason why this space can be seen as a generalization of the formal double
zeta space $\mathcal{D}_{k}$ is that we have the following embedding of
$\mathcal{D}_{k}$ in $\mathcal{E}_{k}$.

######  Proposition 3.2.

For all $k\geq 1$ the following gives a $\mathbb{Q}$-linear map
$\mathcal{D}_{k}\rightarrow\mathcal{E}_{k}$

$\displaystyle Z_{k}$ $\displaystyle\longmapsto
G\genfrac{(}{)}{0.0pt}{}{k}{0}-\delta_{k,2}G\genfrac{(}{)}{0.0pt}{}{2}{0}\,,$
$\displaystyle\quad Z_{k_{1},k_{2}}$ $\displaystyle\longmapsto
G\genfrac{(}{)}{0.0pt}{}{k_{1},k_{2}}{0,0}+\frac{1}{2}\left(\delta_{k_{2},1}G\genfrac{(}{)}{0.0pt}{}{k_{1}}{1}-\delta_{k_{1},1}G\genfrac{(}{)}{0.0pt}{}{k_{2}}{1}+\delta_{k_{1},2}G\genfrac{(}{)}{0.0pt}{}{k_{2}+1}{1}\right)\,,$
$\displaystyle P_{k_{1},k_{2}}$ $\displaystyle\longmapsto
P\genfrac{(}{)}{0.0pt}{}{k_{1},k_{2}}{0,0}+\frac{1}{2}\left(\delta_{k_{1},2}G\genfrac{(}{)}{0.0pt}{}{k_{2}+1}{1}+\delta_{k_{2},2}G\genfrac{(}{)}{0.0pt}{}{k_{1}+1}{1}\right)-\delta_{k_{1}\cdot
k_{2},1}G\genfrac{(}{)}{0.0pt}{}{2}{0}\,.$

###### Proof.

We just need to check that this map is well-defined. This can be done by
applying this map to the defining equation (1.3) of $\mathcal{D}_{k}$ and then
observe that, after rearranging some terms and using
$G\genfrac{(}{)}{0.0pt}{}{2}{0}=G\genfrac{(}{)}{0.0pt}{}{1}{1}$, the result is
exactly (3.1) with $d_{1}=d_{2}=0$. ∎

Due to Proposition 3.2 any relation in the formal double zeta space also gives
a relation in the formal double Eisenstein space. We will use this in the
proof of Theorem 4.1 below. First we will be interested in realizations of the
space $\mathcal{E}_{k}$, by which we again mean elements in
$\operatorname{Hom}_{\mathbb{Q}}(\mathcal{E}_{k},A)$ for some
$\mathbb{Q}$-vector space $A$. Also notice that by Proposition 3.2 any
realization of $\mathcal{E}_{k}$ gives a realization of $\mathcal{D}_{k}$. To
write down explicit realizations for all weight $k$ we will consider
generating series defined as follows

$\displaystyle\mathfrak{G}_{1}\\!\genfrac{(}{)}{0.0pt}{}{X_{1}}{Y_{1}}:=$
$\displaystyle\sum_{\begin{subarray}{c}k_{1}\geq 1\\\ d_{1}\geq
0\end{subarray}}G\genfrac{(}{)}{0.0pt}{}{k_{1}}{d_{1}}X_{1}^{k_{1}-1}\frac{Y_{1}^{d_{1}}}{d_{1}!}\,,\qquad\mathfrak{G}_{2}\\!\genfrac{(}{)}{0.0pt}{}{X_{1},X_{2}}{Y_{1},Y_{2}}:=\sum_{\begin{subarray}{c}k_{1},k_{2}\geq
1\\\ d_{1},d_{2}\geq
0\end{subarray}}G\genfrac{(}{)}{0.0pt}{}{k_{1},k_{2}}{d_{1},d_{2}}X_{1}^{k_{1}-1}X_{2}^{k_{2}-1}\frac{Y_{1}^{d_{1}}}{d_{1}!}\frac{Y_{2}^{d_{2}}}{d_{2}!}\,,$
$\displaystyle\mathfrak{P}\\!\genfrac{(}{)}{0.0pt}{}{X_{1},X_{2}}{Y_{1},Y_{2}}:=$
$\displaystyle\sum_{\begin{subarray}{c}k_{1},k_{2}\geq 1\\\ d_{1},d_{2}\geq
0\end{subarray}}P\genfrac{(}{)}{0.0pt}{}{k_{1},k_{2}}{d_{1},d_{2}}X_{1}^{k_{1}-1}X_{2}^{k_{2}-1}\frac{Y_{1}^{d_{1}}}{d_{1}!}\frac{Y_{2}^{d_{2}}}{d_{2}!}\,.$

With this (3.1) can be written as

(3.2)
$\displaystyle\begin{split}\mathfrak{P}\\!\genfrac{(}{)}{0.0pt}{}{X_{1},X_{2}}{Y_{1},Y_{2}}&=\mathfrak{G}_{2}\\!\genfrac{(}{)}{0.0pt}{}{X_{1},X_{2}}{Y_{1},Y_{2}}+\mathfrak{G}_{2}\\!\genfrac{(}{)}{0.0pt}{}{X_{2},X_{1}}{Y_{2},Y_{1}}+\frac{\mathfrak{G}_{1}\\!\genfrac{(}{)}{0.0pt}{}{X_{1}}{Y_{1}+Y_{2}}-\mathfrak{G}_{1}\\!\genfrac{(}{)}{0.0pt}{}{X_{2}}{Y_{1}+Y_{2}}}{X_{1}-X_{2}}\\\
&=\mathfrak{G}_{2}\\!\genfrac{(}{)}{0.0pt}{}{X_{1}+X_{2},X_{2}}{Y_{1},Y_{2}-Y_{1}}+\mathfrak{G}_{2}\\!\genfrac{(}{)}{0.0pt}{}{X_{1}+X_{2},X_{1}}{Y_{2},Y_{1}-Y_{2}}+\frac{\mathfrak{G}_{1}\\!\genfrac{(}{)}{0.0pt}{}{X_{1}+X_{2}}{Y_{1}}-\mathfrak{G}_{1}\\!\genfrac{(}{)}{0.0pt}{}{X_{1}+X_{2}}{Y_{2}}}{Y_{1}-Y_{2}}\,.\end{split}$

Finding a realization for all $k\geq 1$ of $\mathcal{E}_{k}$ in some space $A$
is therefore equivalent of finding power series with coefficients in $A$
satisfying (3.2). In [BKM] a systematic way of finding power series satisfying
these equations will be presented. In the following we give one particular
example of such a family of power series without giving its precise origin.
For this we define

$\displaystyle\mathfrak{b}_{1}\genfrac{(}{)}{0.0pt}{}{X}{Y}$
$\displaystyle:=-\frac{1}{4}\left(\coth\left(\frac{X}{2}\right)+\coth\left(\frac{Y}{2}\right)\right)+\frac{1}{2}\left(\frac{1}{X}+\frac{1}{Y}\right)$

and with this

$\displaystyle
R^{\ast}_{\mathfrak{b}}\genfrac{(}{)}{0.0pt}{}{X_{1},X_{2}}{Y_{1},Y_{2}}$
$\displaystyle:=\frac{\mathfrak{b}_{1}\genfrac{(}{)}{0.0pt}{}{X_{1}}{Y_{1}+Y_{2}}-\mathfrak{b}_{1}\genfrac{(}{)}{0.0pt}{}{X_{2}}{Y_{1}+Y_{2}}}{X_{1}-X_{2}}\,,\quad
R^{\shuffle}_{\mathfrak{b}}\genfrac{(}{)}{0.0pt}{}{X_{1},X_{2}}{Y_{1},Y_{2}}:=\frac{\mathfrak{b}_{1}\genfrac{(}{)}{0.0pt}{}{X_{1}+X_{2}}{Y_{1}}-\mathfrak{b}_{1}\genfrac{(}{)}{0.0pt}{}{X_{1}+X_{2}}{Y_{2}}}{Y_{1}-Y_{2}}\,,$
$\displaystyle\mathfrak{P}_{\mathfrak{b}}\genfrac{(}{)}{0.0pt}{}{X_{1},X_{2}}{Y_{1},Y_{2}}$
$\displaystyle:=\mathfrak{b}_{1}\genfrac{(}{)}{0.0pt}{}{X_{1}}{Y_{1}}\mathfrak{b}_{1}\genfrac{(}{)}{0.0pt}{}{X_{2}}{Y_{2}}\,,$
$\displaystyle\mathfrak{b}_{2}\genfrac{(}{)}{0.0pt}{}{X_{1},X_{2}}{Y_{1},Y_{2}}$
$\displaystyle:=\frac{1}{3}\mathfrak{P}_{\mathfrak{b}}\genfrac{(}{)}{0.0pt}{}{X_{1},X_{2}}{Y_{1},Y_{2}}+\frac{1}{3}\mathfrak{P}_{\mathfrak{b}}\genfrac{(}{)}{0.0pt}{}{X_{1}-X_{2},X_{2}}{Y_{1},Y_{1}+Y_{2}}$
$\displaystyle-\frac{5}{12}R^{\shuffle}_{\mathfrak{b}}\genfrac{(}{)}{0.0pt}{}{X_{1}-X_{2},X_{2}}{Y_{1},Y_{1}+Y_{2}}-\frac{1}{12}R^{\shuffle}_{\mathfrak{b}}\genfrac{(}{)}{0.0pt}{}{-X_{2},X_{1}}{-Y_{2},Y_{1}}+\frac{1}{4}R^{\shuffle}_{\mathfrak{b}}\genfrac{(}{)}{0.0pt}{}{X_{2}-X_{1},X_{1}}{Y_{2},Y_{1}+Y_{2}}$
$\displaystyle-\frac{5}{12}R^{\ast}_{\mathfrak{b}}\genfrac{(}{)}{0.0pt}{}{X_{1},X_{2}}{Y_{1},Y_{2}}-\frac{1}{12}R^{\ast}_{\mathfrak{b}}\genfrac{(}{)}{0.0pt}{}{X_{2}-X_{1},X_{2}}{-Y_{1},Y_{1}+Y_{2}}+\frac{1}{4}R^{\ast}_{\mathfrak{b}}\genfrac{(}{)}{0.0pt}{}{X_{1}-X_{2},X_{1}}{-Y_{2},Y_{1}+Y_{2}}\,.$

Writing $\coth$ in terms of exponential functions one then checks by direct
calculation that

(3.3)
$\displaystyle\begin{split}\mathfrak{P}_{\mathfrak{b}}\genfrac{(}{)}{0.0pt}{}{X_{1},X_{2}}{Y_{1},Y_{2}}&=\mathfrak{b}_{2}\genfrac{(}{)}{0.0pt}{}{X_{1},X_{2}}{Y_{1},Y_{2}}+\mathfrak{b}_{2}\genfrac{(}{)}{0.0pt}{}{X_{2},X_{1}}{Y_{2},Y_{1}}+\frac{\mathfrak{b}_{1}\genfrac{(}{)}{0.0pt}{}{X_{1}}{Y_{1}+Y_{2}}-\mathfrak{b}_{1}\genfrac{(}{)}{0.0pt}{}{X_{2}}{Y_{1}+Y_{2}}}{X_{1}-X_{2}}\\\
&=\mathfrak{b}_{2}\genfrac{(}{)}{0.0pt}{}{X_{1}+X_{2},X_{2}}{Y_{1},Y_{2}-Y_{1}}+\mathfrak{b}_{2}\genfrac{(}{)}{0.0pt}{}{X_{1}+X_{2},X_{1}}{Y_{2},Y_{1}-Y_{2}}+\frac{\mathfrak{b}_{1}\genfrac{(}{)}{0.0pt}{}{X_{1}+X_{2}}{Y_{1}}-\mathfrak{b}_{1}\genfrac{(}{)}{0.0pt}{}{X_{1}+X_{2}}{Y_{2}}}{Y_{1}-Y_{2}}\,,\end{split}$

i.e. this gives a realization of $\mathcal{E}_{k}$ in $\mathbb{Q}$ for any
$k\geq 1$ by taking the coefficients of the powers series
$\mathfrak{b}_{1},\mathfrak{b}_{2}$ and $\mathfrak{P}_{\mathfrak{b}}$ for the
images of
$G\genfrac{(}{)}{0.0pt}{}{k}{d},G\genfrac{(}{)}{0.0pt}{}{k_{1},k_{2}}{d_{1},d_{2}}$
and $P\genfrac{(}{)}{0.0pt}{}{k_{1},k_{2}}{d_{1},d_{2}}$ respectively. In
depth one this realization satisfies
$G\genfrac{(}{)}{0.0pt}{}{k}{0}\mapsto-\frac{B_{k}}{2k!}$ for $k>1$ and
together with Proposition 3.2 this gives a realization of $\mathcal{D}_{k}$ in
$\mathbb{Q}$ which is called the _Bernoulli realization_ in [GKZ]. The above
realization will give the constant term of the Eisenstein realization which we
will introduce now. For this we first define the generating series of the
$q$-series $g$

$\displaystyle\mathfrak{g}_{1}\genfrac{(}{)}{0.0pt}{}{X}{Y}:=\sum_{\begin{subarray}{c}k\geq
1\\\ d\geq
0\end{subarray}}g\genfrac{(}{)}{0.0pt}{}{k}{d}X^{k-1}\frac{Y^{d}}{d!}\,,\quad\mathfrak{g}_{2}\genfrac{(}{)}{0.0pt}{}{X_{1},X_{2}}{Y_{1},Y_{2}}:=\sum_{\begin{subarray}{c}k_{1},k_{2}\geq
1\\\ d_{1},d_{2}\geq
0\end{subarray}}g\genfrac{(}{)}{0.0pt}{}{k_{1},k_{2}}{d_{1},d_{2}}X_{1}^{k_{1}-1}X_{2}^{k_{2}-1}\frac{Y_{1}^{d_{1}}}{d_{1}!}\frac{Y_{2}^{d_{2}}}{d_{2}!}\,.$

Using these generating series the stuffle product analogue (Proposition 2.7),
the partition relation (Proposition 2.8) and the shuffle product analogue
(Proposition 2.10) can be expressed as follows.

######  Lemma 3.3.

The generating series $\mathfrak{g}_{1}$ and $\mathfrak{g}_{2}$ satisfy the
following relations

  1. i)

Stuffle product analogue:

$\displaystyle\begin{split}\mathfrak{g}_{1}\genfrac{(}{)}{0.0pt}{}{X_{1}}{Y_{1}}\mathfrak{g}_{1}\genfrac{(}{)}{0.0pt}{}{X_{2}}{Y_{2}}=&\,\mathfrak{g}_{2}\genfrac{(}{)}{0.0pt}{}{X_{1},X_{2}}{Y_{1},Y_{2}}+\mathfrak{g}_{2}\genfrac{(}{)}{0.0pt}{}{X_{2},X_{1}}{Y_{2},Y_{1}}+\frac{1}{X_{1}-X_{2}}\left(\mathfrak{g}_{1}\genfrac{(}{)}{0.0pt}{}{X_{1}}{Y_{1}+Y_{2}}-\mathfrak{g}_{1}\genfrac{(}{)}{0.0pt}{}{X_{2}}{Y_{1}+Y_{2}}\right)\\\
&+\Big{(}\mathfrak{b}_{1}\genfrac{(}{)}{0.0pt}{}{X_{2}-X_{1}}{Y_{1}+Y_{2}}-\mathfrak{b}_{1}\genfrac{(}{)}{0.0pt}{}{X_{1}-X_{2}}{Y_{1}+Y_{2}}\Big{)}\left(\mathfrak{g}_{1}\genfrac{(}{)}{0.0pt}{}{X_{1}}{Y_{1}+Y_{2}}-\mathfrak{g}_{1}\genfrac{(}{)}{0.0pt}{}{X_{2}}{Y_{1}+Y_{2}}\right)\\\
&-\frac{1}{2}\left(\mathfrak{g}_{1}\genfrac{(}{)}{0.0pt}{}{X_{1}}{Y_{1}+Y_{2}}+\mathfrak{g}_{1}\genfrac{(}{)}{0.0pt}{}{X_{2}}{Y_{1}+Y_{2}}\right)\,.\end{split}$

  2. ii)

Partition relation:

$\displaystyle\mathfrak{g}_{1}\genfrac{(}{)}{0.0pt}{}{X_{1}}{Y_{1}}=\mathfrak{g}_{1}\genfrac{(}{)}{0.0pt}{}{Y_{1}}{X_{1}}\,,\qquad\mathfrak{g}_{2}\genfrac{(}{)}{0.0pt}{}{X_{1},X_{2}}{Y_{1},Y_{2}}=\mathfrak{g}_{2}\genfrac{(}{)}{0.0pt}{}{Y_{1}+Y_{2},Y_{1}}{X_{2},X_{1}-X_{2}}\,.$

  3. iii)

Shuffle product analogue:

$\displaystyle\begin{split}\mathfrak{g}_{1}\genfrac{(}{)}{0.0pt}{}{X_{1}}{Y_{1}}\mathfrak{g}_{1}\genfrac{(}{)}{0.0pt}{}{X_{2}}{Y_{2}}=&\,\mathfrak{g}_{2}\genfrac{(}{)}{0.0pt}{}{X_{1}+X_{2},X_{1}}{Y_{2},Y_{1}-Y_{2}}+\mathfrak{g}_{2}\genfrac{(}{)}{0.0pt}{}{X_{1}+X_{2},X_{2}}{Y_{1},Y_{2}-Y_{1}}+\frac{1}{Y_{1}-Y_{2}}\left(\mathfrak{g}_{1}\genfrac{(}{)}{0.0pt}{}{X_{1}+X_{2}}{Y_{1}}-\mathfrak{g}_{1}\genfrac{(}{)}{0.0pt}{}{X_{1}+X_{2}}{Y_{2}}\right)\\\
&+\Big{(}\mathfrak{b}_{1}\genfrac{(}{)}{0.0pt}{}{Y_{2}-Y_{1}}{X_{1}+X_{2}}-\mathfrak{b}_{1}\genfrac{(}{)}{0.0pt}{}{Y_{1}-Y_{2}}{X_{1}+X_{2}}\Big{)}\
\left(\mathfrak{g}_{1}\genfrac{(}{)}{0.0pt}{}{X_{1}+X_{2}}{Y_{1}}-\mathfrak{g}_{1}\genfrac{(}{)}{0.0pt}{}{X_{1}+X_{2}}{Y_{2}}\right)\\\
&-\frac{1}{2}\left(\mathfrak{g}_{1}\genfrac{(}{)}{0.0pt}{}{X_{1}+X_{2}}{Y_{1}}+\mathfrak{g}_{1}\genfrac{(}{)}{0.0pt}{}{X_{1}+X_{2}}{Y_{2}}\right)\,.\end{split}$

###### Proof.

These are exactly the statements of Proposition 2.7, 2.8 and 2.10 written down
in terms of generating series. ∎

Using Lemma 3.3 together with (3.3) one can construct a realization of
$\mathcal{E}_{k}$ for all $k\geq 1$ with
$G\genfrac{(}{)}{0.0pt}{}{k}{0}\mapsto-\frac{B_{k}}{2k!}+g(k)=\widetilde{G}_{k}$
for $k>1$ such that
$P\genfrac{(}{)}{0.0pt}{}{k_{1},k_{2}}{0,0}\mapsto\widetilde{G}_{k_{1}}\widetilde{G}_{k_{2}}$.
The hard part of this is to find the correct image in depth two. There exist a
general construction behind this, which is inspired by the Fourier expansion
of double Eisenstein series, and which will be explained in detail in [BB].

######  Theorem 3.4.

The power series

$\displaystyle\mathfrak{E}_{1}\genfrac{(}{)}{0.0pt}{}{X_{1}}{Y_{1}}$
$\displaystyle:=\mathfrak{b}_{1}\genfrac{(}{)}{0.0pt}{}{X_{1}}{Y_{1}}+\mathfrak{g}_{1}\genfrac{(}{)}{0.0pt}{}{X_{1}}{Y_{1}}\,,$
$\displaystyle\mathfrak{E}_{2}\genfrac{(}{)}{0.0pt}{}{X_{1},X_{2}}{Y_{1},Y_{2}}$
$\displaystyle:=\mathfrak{b}_{2}\genfrac{(}{)}{0.0pt}{}{X_{1},X_{2}}{Y_{1},Y_{2}}-\mathfrak{b}_{1}\genfrac{(}{)}{0.0pt}{}{X_{1}-X_{2}}{Y_{2}}\mathfrak{g}_{1}\genfrac{(}{)}{0.0pt}{}{X_{1}}{Y_{1}+Y_{2}}-\frac{1}{2}\mathfrak{g}_{1}\genfrac{(}{)}{0.0pt}{}{X_{1}}{Y_{1}+Y_{2}}$
$\displaystyle+\mathfrak{b}_{1}\genfrac{(}{)}{0.0pt}{}{X_{2}}{Y_{2}}\mathfrak{g}_{1}\genfrac{(}{)}{0.0pt}{}{X_{1}}{Y_{1}}+\mathfrak{b}_{1}\genfrac{(}{)}{0.0pt}{}{X_{1}-X_{2}}{Y_{1}}\mathfrak{g}_{1}\genfrac{(}{)}{0.0pt}{}{X_{2}}{Y_{1}+Y_{2}}+\mathfrak{g}_{2}\genfrac{(}{)}{0.0pt}{}{X_{1},X_{2}}{Y_{1},Y_{2}}$

and
$\mathfrak{P}_{\mathfrak{E}}\genfrac{(}{)}{0.0pt}{}{X_{1},X_{2}}{Y_{1},Y_{2}}=\mathfrak{E}_{1}\\!\genfrac{(}{)}{0.0pt}{}{X_{1}}{Y_{1}}\mathfrak{E}_{1}\\!\genfrac{(}{)}{0.0pt}{}{X_{2}}{Y_{2}}$
satisfy

$\displaystyle\begin{split}\mathfrak{P}_{\mathfrak{E}}\genfrac{(}{)}{0.0pt}{}{X_{1},X_{2}}{Y_{1},Y_{2}}&=\mathfrak{E}_{2}\genfrac{(}{)}{0.0pt}{}{X_{1},X_{2}}{Y_{1},Y_{2}}+\mathfrak{E}_{2}\genfrac{(}{)}{0.0pt}{}{X_{2},X_{1}}{Y_{2},Y_{1}}+\frac{\mathfrak{E}_{1}\genfrac{(}{)}{0.0pt}{}{X_{1}}{Y_{1}+Y_{2}}-\mathfrak{E}_{1}\genfrac{(}{)}{0.0pt}{}{X_{2}}{Y_{1}+Y_{2}}}{X_{1}-X_{2}}\\\
&=\mathfrak{E}_{2}\genfrac{(}{)}{0.0pt}{}{X_{1}+X_{2},X_{2}}{Y_{1},Y_{2}-Y_{1}}+\mathfrak{E}_{2}\genfrac{(}{)}{0.0pt}{}{X_{1}+X_{2},X_{1}}{Y_{2},Y_{1}-Y_{2}}+\frac{\mathfrak{E}_{1}\genfrac{(}{)}{0.0pt}{}{X_{1}+X_{2}}{Y_{1}}-\mathfrak{E}_{1}\genfrac{(}{)}{0.0pt}{}{X_{1}+X_{2}}{Y_{2}}}{Y_{1}-Y_{2}}\,.\end{split}$

###### Proof.

This follows by using Lemma 3.3 together with (3.3). ∎

From Theorem 3.4 we obtain a realization of $\mathcal{E}_{k}$ in
$\mathbb{Q}[[q]]$ for any $k\geq 1$ by taking the coefficients of the powers
series $\mathfrak{E}_{1},\mathfrak{E}_{2}$ and $\mathfrak{P}_{\mathfrak{E}}$
for the images of
$G\genfrac{(}{)}{0.0pt}{}{k}{d},G\genfrac{(}{)}{0.0pt}{}{k_{1},k_{2}}{d_{1},d_{2}}$
and $P\genfrac{(}{)}{0.0pt}{}{k_{1},k_{2}}{d_{1},d_{2}}$ respectively. We call
this realization the _Eisenstein realization_. In particular this realization
satisfies

(3.4)
$\displaystyle\begin{split}G\genfrac{(}{)}{0.0pt}{}{k}{d}&\longmapsto\frac{(k-d-1)!}{(k-1)!}\left(q\frac{d}{dq}\right)^{d}\widetilde{G}_{k-d},\qquad(k>d\geq
0)\\\
P\genfrac{(}{)}{0.0pt}{}{k_{1},k_{2}}{0,0}&\longmapsto\widetilde{G}_{k_{1}}\widetilde{G}_{k_{2}},\qquad(k_{1},k_{2}\geq
1)\,,\end{split}$

where $\widetilde{G}_{k}$ is defined by (2.1) for $k\geq 2$ and
$\widetilde{G}_{1}:=g(1)$.

######  Remark 3.5.

We call the coefficients of $\mathfrak{E}_{1}$ and $\mathfrak{E}_{2}$
_combinatorial multiple Eisenstein series_. In [BB] these objects will be
considered for higher depths and the construction in Theorem 3.4 will be
explained in detail.

##  4\. Application

In the formal double Eisenstein space a lot explicit relations can be proven
by using the defining relations (3.1). A lot of these relations will be
explained in [BKM] and we will just mention the following Theorem which
follows from a result proven in [B2] for the formal double zeta space.

######  Theorem 4.1.

For all $k_{1},k_{2}\geq 1$ with $k=k_{1}+k_{2}\geq 4$ even we have

(4.1)
$\displaystyle\begin{split}\frac{1}{2}\left(\binom{k_{1}+k_{2}}{k_{2}}-(-1)^{k_{1}}\right)G\genfrac{(}{)}{0.0pt}{}{k}{0}=&\sum_{\begin{subarray}{c}j=2\\\
j\text{even}\end{subarray}}^{k-2}\left(\binom{k-j-1}{k_{1}-1}+\binom{k-j-1}{k_{2}-1}-\delta_{j,k_{1}}\right)P\genfrac{(}{)}{0.0pt}{}{j,k-j}{0,0}\\\
&+\frac{1}{2}\left(\binom{k-3}{k_{1}-1}+\binom{k-3}{k_{2}-1}+\delta_{k_{1},1}+\delta_{k_{2},1}\right)G\genfrac{(}{)}{0.0pt}{}{k-1}{1}\,.\end{split}$

###### Proof.

By [B2, Theorem 4.9] the following relation in the formal double zeta space
holds

$\displaystyle\frac{1}{2}\left(\binom{k_{1}+k_{2}}{k_{2}}-(-1)^{k_{1}}\right)Z_{k}=\sum_{\begin{subarray}{c}j=2\\\
j\text{even}\end{subarray}}^{k-2}\left(\binom{k-j-1}{k_{1}-1}+\binom{k-j-1}{k_{2}-1}-\delta_{j,k_{1}}\right)P_{j,k-j}\,.$

Applying the linear map from Proposition 3.2 to this gives exactly (4.1). ∎

In [BKM] a generalization of Theorem 4.1 will be presented. This theorem has
nice special cases which we will give now.

######  Corollary 4.2.

  1. i)

For even $k\geq 4$ we have

$\displaystyle
G\genfrac{(}{)}{0.0pt}{}{k-1}{1}=\frac{k+1}{2}G\genfrac{(}{)}{0.0pt}{}{k}{0}+\sum_{\begin{subarray}{c}k_{1}+k_{2}=k\\\
k_{1},k_{2}\geq 2\text{
even}\end{subarray}}P\genfrac{(}{)}{0.0pt}{}{k_{1},k_{2}}{0,0}\,.$

  2. ii)

For all even $k\geq 6$ we have

$\displaystyle\frac{(k+1)(k-1)(k-6)}{12}G\genfrac{(}{)}{0.0pt}{}{k}{0}=\sum_{\begin{subarray}{c}k_{1}+k_{2}=k\\\
k_{1},k_{2}\geq 4\text{
even}\end{subarray}}(k_{1}-1)(k_{2}-1)\,P\genfrac{(}{)}{0.0pt}{}{k_{1},k_{2}}{0,0}\,.$

###### Proof.

Part i) is the $k_{1}=1$ case of Theorem 4.1. Part ii) follows by considering
$k\\!-\\!3$-times the $(k_{1},k_{2})=(k-2,2)$ case and then subtracting
$2$-times the $(k_{1},k_{2})=(k-3,3)$ case. ∎

######  Example 4.3.

  1. i)

We have

$\displaystyle G\genfrac{(}{)}{0.0pt}{}{8}{0}=\frac{6}{7}G_{4}^{2},\,\quad
G\genfrac{(}{)}{0.0pt}{}{10}{0}=\frac{10}{11}P\genfrac{(}{)}{0.0pt}{}{4,6}{0,0}\,.$

  2. ii)

Analogues of the Ramanujan differential equations are satisfied

$\displaystyle 2G\genfrac{(}{)}{0.0pt}{}{3}{1}$
$\displaystyle=5G\genfrac{(}{)}{0.0pt}{}{4}{0}-2P\genfrac{(}{)}{0.0pt}{}{2,2}{0,0}\,,\qquad
4G\genfrac{(}{)}{0.0pt}{}{5}{1}=8G\genfrac{(}{)}{0.0pt}{}{6}{0}-14P\genfrac{(}{)}{0.0pt}{}{2,4}{0,0}\,,$
$\displaystyle 6G\genfrac{(}{)}{0.0pt}{}{7}{1}$
$\displaystyle=\frac{120}{7}P\genfrac{(}{)}{0.0pt}{}{4,4}{0,0}-12P\genfrac{(}{)}{0.0pt}{}{2,6}{0,0}\,.$

Applying the Eisenstein realization (3.4) we get the well-known relations
$\widetilde{G}_{8}=\frac{6}{7}\widetilde{G}_{4}^{2}$,
$\widetilde{G}_{10}=\frac{10}{11}\widetilde{G}_{4}\widetilde{G}_{6}$ as well
as the Ramanujan differential equations

$\displaystyle
q\frac{d}{dq}\widetilde{G}_{2}=5\widetilde{G}_{4}-2\widetilde{G}_{2}^{2}\,,\quad
q\frac{d}{dq}\widetilde{G}_{4}=8\widetilde{G}_{6}-14\widetilde{G}_{2}\widetilde{G}_{4}\,,\quad
q\frac{d}{dq}\widetilde{G}_{6}=\frac{120}{7}\widetilde{G}_{4}^{2}-12\widetilde{G}_{2}\widetilde{G}_{6}\,.$

######  Remark 4.4.

  1. i)

These proofs of identities among quasi-modular forms are purely combinatorial
and are similar to those done in [S]. Applying the Eisenstein realization to
Theorem 3.4 can also be compared to the relations proven in [HST, Theorem 1].

  2. ii)

In [BIM] everything done here in depth two will be generalized to arbitrary
depths by introducing the algebra of _formal multiple Eisenstein series_. In
this algebra, one can also define a subalgebra of _formal (quasi-)modular
forms_ for which (almost) all standard relations known for modular forms can
be proven on a formal level.

## References

  * [B1] H. Bachmann: The algebra of bi-brackets and regularized multiple Eisenstein series, J. Number Theory, 200 (2019), 260–294.
  * [B2] H. Bachmann: Multiple zeta values and modular forms, Lecture notes (Ver. 5.4), Nagoya University 2020. available online at https://www.henrikbachmann.com/mzv2020.html.
  * [BB] H. Bachmann, A. Burmester: Combinatorial multiple Eisenstein series, In preparation.
  * [BBK] H. Bachmann, A. Burmester, U. Kühn: Relations satisfied by double q-zeta values, RIMS Kokyuroku No.2160 (2020), p. 219–230.
  * [BK] H. Bachmann, U. Kühn: The algebra of generating functions for multiple divisor sums and applications to multiple zeta values, Ramanujan J., 40 (2016), 605–648.
  * [BIM] H. Bachmann, J.W. van-Ittersum, N. Matthes: Formal multiple Eisenstein series, In preparation.
  * [BKM] H. Bachmann, U. Kühn, N. Matthes: Realizations of the formal double Eisenstein space, In preparation.
  * [GKZ] H. Gangl, M.Kaneko, D. Zagier: Double zeta values and modular forms, in ”Automorphic forms and zeta functions” World Sci. Publ., Hackensack, NJ (2006), 71–106.
  * [HST] M. Hirose, N. Sato and K. Tasaka: Eisenstein series identities based on partial fraction decomposition, Ramanujan J., 38(3) (2015), 455–463.
  * [S] N.P. Skoruppa: A Quick Combinatorial Proof of Eisenstein Series Identities, J. Number Theory, 43(1) (1993), 68–73.