Combinatorial multiple Eisenstein series

We construct a family of q-series with rational coefficients satisfying a variant of the extended double shuffle equations, which are a lift of a given Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Q}$$\end{document}-valued solution of the extended double shuffle equations. We call these q-series combinatorial (bi-)multiple Eisenstein series, and in depth one they coincide with (classical) Eisenstein series. Combinatorial multiple Eisenstein series can be seen as an interpolation between the given Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Q}$$\end{document}-valued solution of the extended double shuffle equations (as q→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\rightarrow 0$$\end{document}) and multiple zeta values (as q→1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\rightarrow 1$$\end{document}). In particular, they are q-analogues of multiple zeta values closely related to modular forms. Their definition is inspired by the Fourier expansion of multiple Eisenstein series introduced by Gangl-Kaneko-Zagier. Our explicit construction is done on the level of their generating series, which we show to be a so-called symmetril and swap invariant bimould.


Introduction
Multiple zeta values are defined for r ě 1 and k 1 ě 2, k 2 , . . ., k r ě 1 by ζpk 1 , . . ., k r q " ÿ m 1 ą¨¨¨ąmrą0 1 m k 1 1 ¨¨¨m kr r . (1.1) We call the number k 1 `¨¨¨`k r its weight and r its depth.By Z, we denote the Q-algebra of all multiple zeta values.There are two ways of expressing the product of multiple zeta values, and both can be written in terms of quasi-shuffle products ( [H]).The relations obtained from the two product expressions, together with some regularization process, are referred to as the extended double shuffle relations of multiple zeta values ( [IKZ]).Conjecturally these give all algebraic relations among multiple zeta values.Multiple zeta values have various different connections to modular forms.For example, in the case r " 1 multiple zeta values are the Riemann zeta values, which also appear as the constant term of Eisenstein series.
In [GKZ] the authors defined double Eisenstein series, which have double zeta values ((1.1) in the case r " 2) as their constant terms, and which in some sense give a natural depth two version of Eisenstein series.This raised the question if these objects also satisfy some of the extended double shuffle relations.Partial answers for this were given in [GKZ] and in arbitrary depths for so-called multiple Eisenstein series in [B1], [B2] and [BT].In this work, we present a new approach and lift Eisenstein series with rational coefficients in a purely combinatorial 1 way to q-series which we call combinatorial multiple Eisenstein series.This provides a new framework for relating modular forms and multiple zeta values.We discuss the relations satisfied by these q-series and give an interpretation of them as a variant of the extended double shuffle relations.We will first recall the extended double shuffle relations for multiple zeta values before explaining how the combinatorial multiple Eisenstein series fit into the picture.Consider the alphabet L z " tz k | k ě 1u and let H 1 " QxL z y be the free algebra over L z .Define a product on QL z by z i ˛zj " z i`j for all i, j ě 1.The corresponding quasi-shuffle product (3.1)˚" ˚˛is usually called harmonic or stuffle product.Let H 0 be the subalgebra of H 1 generated by all words starting not in z 1 .Due to the usual power series multiplication, the linear map2 defined on the generators by ζ : gives an algebra homomorphism from pH 0 , ˚q to Z.This homomorphism can be extended to a homomorphism ζ ˚: H 1 Ñ Z, so we obtain elements ζ ˚pk 1 , . . ., k r q P R for all k 1 , . . ., k r ě 1 called the stuffle regularized multiple zeta values (see [IKZ]).In the case k 1 ě 2 these coincide with the multiple zeta values (1.1) and they are uniquely determined by this property together with ζ ˚p1q " 0 and the fact that the Q-linear map ζ ˚: H 1 Ñ Z defined on the generators by z k 1 . . .z kr Þ Ñ ζ ˚pk 1 , . . ., k r q is an algebra homomorphism from pH 1 , ˚q to Z.
Next, consider the alphabet given by the two letters L xy " tx, yu and write H " QxL xy y.Define the product a ˛b " 0 for all a, b P QL xy , then the corresponding quasi-shuffle product ˚˛is the shuffle product, denoted by ¡.Via the identification z k " x k´1 y we can view H 1 and H 0 as subalgebras of the shuffle algebra pH, ¡q, we have H 1 " Q1 `Hy and H 0 " Q1 `xHy.
Due to the iterated integral expression of multiple zeta values, one obtains that the map (1.2) gives an algebra homomorphism from pH 0 , ¡q to Z.There is also a unique extension of the map ζ to an algebra homomorphism ζ ¡ : pH 1 , ¡q Ñ Z given by shuffle regularized multiple zeta values and satisfying ζ ¡ p1q " 0. These two regularizations differ and their difference can be described explicitly (see [IKZ,Theorem 1]).For example, we have in depth two for all k 1 , k 2 ě 1 where δ denotes the Kronecker delta.We call these equations obtained by comparing products of shuffle-and stuffle-regularized multiple zeta values the extended double shuffle equations (see Definition 3.5 for a precise definition in terms of generating series).
Beside the multiple zeta values there are other objects satisfying the extended double shuffle equations.In particular, it is known that there exist (non-trivial) rational 3 solutions to the extended double shuffle equations, i.e. numbers βpk 1 , . . ., k r q P Q for k 1 ě 2, k 2 , . . ., k r ě 1 and corresponding stuffle and shuffle regularized maps β ˚, β ¡ : H 1 Ñ Q.In this article, we will focus on the stuffle regularized objects and thus we write by abuse of notation β " β ˚.
We will restrict to rational solutions, which in depth one for even k ě 2 are given by βpkq " ζpkq p2πiq k " ´Bk 2k! (1.4) and for odd k ě 1 by βpkq " 0. These rational numbers also appear as the constant terms in the Fourier expansion of the Eisenstein series, defined for k ě 1 by Gpkq " βpkq `1 pk ´1q! ÿ d,mě1 d k´1 q md P Q q . (1.5) For even k ě 4 these are, when viewed as functions in τ P H " tτ P C | Impτ q ą 0u with q " e 2πiτ , modular forms of weight k for the full modular group.In our context, they can also be seen as interpolations between ζpkq and βpkq, i.e. the depth one objects of the two solutions of the extended double shuffle equations mentioned above.More precisely, we have lim qÑ0 Gpkq " βpkq and lim qÑ1 p1´qq k Gpkq " ζpkq, where the latter is a consequence of (2.4).
The construction of the combinatorial multiple Eisenstein series depends of the choice of the rational solution to the extended double shuffle equations β, though most of their property will be independent of this choice as we already see in (1.6).Moreover, the combinatorial multiple Eisenstein series can also be viewed as a map G : H 1 Ñ Q q satisfying for w, v P H 1 an analogue of the extended double shuffle equations.For example as an analogue of (1.3) we have for k 1 , k 2 ě 1 (Proposition 6.7) where the q-series R G pk 1 , k 2 q is given by Observe that we have lim qÑ0 R G pk 1 , k 2 q " δ k 1 `k2 ,2 βp2q and lim qÑ1 p1 ´qq k 1 `k2 R G pk 1 , k 2 q " δ k 1 `k2 ,2 ζp2q.In particular, the formula (1.7) gives an explicit expression for q d dq Gpkq in terms of combinatorial double Eisenstein series by choosing k 2 " 2. This actually works for arbitrary depths, for any w P H 1 we will see that (Corollary 6.31) Gpwq " Gpz 2 ˚w ´z2 ¡ wq .This is a nice example for the fact that the derivatives are an obstacle for the combinatorial multiple Eisenstein series satisfying the extended double shuffle relations.The expression Gpz 2 ˚w ´z2 ¡ wq does not vanish in general, but it is exactly given by a derivative.So in particular, its constant term (and also its limit for q Ñ 1) indeed vanishes.
In order to deal with derivatives and to include them into the algebraic setup, we will consider objects depending on double indices.More precisely, we will introduce combinatorial bi-multiple Eisenstein series G `k1 ,...,kr d 1 ,...,dr ˘P Q q defined for k 1 , . . ., k r ě 1 and d 1 , . . ., d r ě 0. The sum k 1 `¨¨¨`k r `d1 `¨¨¨`d r will be called its weight.The combinatorial multiple Eisenstein series are given in the special case Gpk 1 , . . ., k r q " G ˆk1 , . . ., k r 0, . . ., 0
In some special cases, the combinatorial bi-multiple Eisenstein series are modular (Proposition 6.13) and the same holds for some linear combinations (Proposition 6.19).But in general, the combinatorial bi-multiple Eisenstein series do not satisfy the modularity condition and it is not clear which linear combinations of them do.
In contrast to the case of multiple zeta values, we will not describe the relations of the combinatorial bi-multiple Eisenstein series in terms of two different product expressions.More explicitly, we will consider a bi-version of the stuffle product and a second family of relations will be the invariance under a certain involution.This involution has a natural origin coming from the theory of partitions ([B1], [B2], [BI], [Bri]) and it can be described nicely in terms of generating series.Therefore, we will work entirely with generating series for the construction of the combinatorial bi-multiple Eisenstein series.For r ě 1 these will be denoted by and in general a collection of such generating series for all r will be called a bimould (Definition 3.7).To describe the bi-analogue of the stuffle product we consider the alphabet Then we call the bimould G symmetril (Definition 3.8) if the linear map, defined on the generators by gives an algebra homomorphism from pQxL bi z y, ˚q to Q q .The bimould G will be called swap invariant (Definition 3.10), if it satisfies for all r ě 1 the functional equation which implies linear relations among combinatorial bi-multiple Eisenstein series in homogeneous weight.The main result of this work will be the following.
Theorem (Theorem 6.5, Proposition 6.17).Let β be a Q-valued solution to the extended double shuffle equations, which is in depth one given by (4.1).Then there exists a Q qvalued bimould G, which is symmetril, swap invariant and whose coefficients in depth one are the Eisenstein series (1.5), i.e.
The coefficients of the bimould G interpolate between (stuffle regularized) multiple zeta values and the given Q-valued solution β, i.e., they satisfy (1.6).
From this theorem we get that the combinatorial multiple Eisenstein series Gpk 1 , . . ., k r q satisfy the stuffle product formula.By combining symmetrility and swap invariance of G, we get that the combinatorial (bi-)multiple Eisenstein series also satisfy an analogue of the shuffle product formula.We will work this out in depth two in Proposition 6.7.
The construction of the bimould G is inspired by the calculation of the Fourier expansion of the multiple Eisenstein series G introduced by Gangl-Kaneko-Zagier ([B2], [GKZ]).We will recall this calculation (Theorem 2.1) in Section 2. The following diagram provides a rough overview of how the building blocks of our constructions (right-hand side) are related to the classical building blocks of multiple Eisenstein series (left-hand side).In particular, the bimould G will be constructed out of four bimoulds b, g ˚, L m and g, whose constructions are all inspired by the corresponding objects/statements in Section 2. We will show that the bimoulds g ˚and b are symmetril (Proposition 6.22, 4.7), hence the same holds for the bimould G (by Proposition 3.9).On the other hand, the bimould G is a sum of swap invariant bimoulds G j (Theorem 6.26, Proposition 6.27), thus G is also swap invariant.
By (1.6) combinatorial multiple Eisenstein series can also be interpreted as q-analogues of multiple zeta values.Our notion of weight is compatible with the weight of quasi-modular forms, and both product expressions of the combinatorial bi-multiple Eisenstein series (given in Proposition 6.7 for depth two) are homogeneous in weight.As far as the authors know, combinatorial multiple Eisenstein series provide the first model of q-analogues of multiple zeta values with this property.In particular, this might give a positive answer to a question raised by Okounkov in [O], since the space qMZV introduced there is exactly spanned by all Gpk 1 , . . ., k r q with k 1 , . . ., k r ě 2.Moreover, we show (Proposition 6.15) that the combinatorial bi-multiple Eisenstein series span the space of q-analogues of multiple zeta values Z q considered5 in [B1], [B2], [BI] and [BK2].
Conjecturally all algebraic relations among combinatorial bi-multiple Eisenstein are consequences of combining the symmetrility and the swap invariance (Remark 6.11).Since these relations are all in homogeneous weight, we, in particular, expect that the combinatorial bi-multiple Eisenstein are graded by weight.
In [BIM] the authors introduce the algebra of formal multiple Eisenstein series, which is generated by formal symbols modulo the relations coming from combining the symmetrility and swap invariance.In this algebra one can also define a projection to the space of formal multiple zeta values, which can be seen as a formal version of (1.6).Further it is shown, that the sl 2 -action from quasi-modular forms can be extended to this algebra.By the above mentioned conjecture, the algebra of combinatorial bi-multiple Eisenstein series G bi (Definition 6.10) should be isomorphic to the algebra of formal multiple Eisenstein series and therefore G bi should also be an sl 2 -algebra.
A similar formal algebraic approach is used independently in the thesis of the second named author [Bu].Here another quasi-shuffle algebra is considered together with an involution, which is of a simpler shape than the operator swap.It is shown in [Bu2,Theorem 7.10] that this weight-graded algebra is isomorphic to the weight-graded algebra of formal multiple Eisenstein series.The description in terms of this other quasi-shuffle algebra seems to be a good choice to proceed as in [R], which means giving a generalization of the pro-unipotent affine group scheme DM and the double shuffle Lie algebra dm 0 .
Finally we remark that the name of the combinatorial multiple Eisenstein series was inspired by the combinatorial double Eisenstein series Z k 1 ,k 2 introduced in [GKZ,(17)].These differ slightly to our Gpk 1 , k 2 q, but can be related by using [BKM,Proposition 2.5] and adding the constant term βpk 1 , k 2 q.Combinatorial multiple Eisenstein series might also have a, yet to understood, connection to iterated integrals of quasimodular forms ([M]).

Multiple Eisenstein series
In this section, we recall multiple Eisenstein series and the calculation of their Fourier expansion.Details can be found in [B1], [B2], and [BT].This will give a motivation and an explanation for our construction of combinatorial multiple Eisenstein series in Section 6.
For k 1 ě 3, k 2 , . . ., k r ě 2 and τ P H the multiple Eisenstein series are defined by G k 1 ,...,kr pτ q :" ÿ where the order ą on the lattice Zτ `Z is defined by ..,kr pτ `1q " G k 1 ,...,kr pτ q the multiple Eisenstein series possess a Fourier expansion, i.e. an expansion in q " e 2πiτ , which was calculated in [GKZ] for the r " 2 case and for arbitrary depth by the first author ( [B2]).In depth one we have for k ě 3 For even k ě 4 these are just the classical Eisenstein series, which are modular forms for the full modular group.When k is even, these differ from the Eisenstein series (1.5) defined in the introduction just by a factor of p2πiq k .We refer to Ψ k pτ q as the monotangent function ( [Bo]), which satisfies for k ě 2 the Lipschitz formula (2.2) This gives d k´1 q md ": ζpkq `p´2πiq k gpkq .
Here the gpkq are the generating series of the divisor-sums and for higher depths multiple versions of these q-series appear, which are defined for k 1 , . . .k r ě 1 by (2.3) These q-series were studied in detail in [B2], [BK] and they can be seen as q-analogues of multiple zeta values since one can show that for k 1 ě 2 lim qÑ1 p1 ´qq k 1 `¨¨¨`kr gpk 1 , . . ., k r q " ζpk 1 , . . ., k r q . (2.4) In the Fourier expansion of (multiple) Eisenstein series, the q-series g always appear together with a power of ´2πi and therefore we set for k 1 , . . ., k r ě 1 ĝpk 1 , . . ., k r q :" p´2πiq k 1 `¨¨¨`kr gpk 1 , . . ., k r q P Qrπis q .
A multiple version of G k pτ q " ζpkq `ĝpkq is given by the following.
We will sketch the proof of Theorem 2.1 in the following and then give an explicit example at the end of the section.First, observe that for k 1 , . . ., k r ě 2 we have by the Lipschitz formula (2.2), that the q-series ĝ can be written as an ordered sum over monotangent functions ĝpk 1 , . . ., k r q " ÿ m 1 ą¨¨¨ąmrą0 Ψ k 1 pm 1 τ q ¨¨¨Ψ kr pm r τ q . (2.5) In general the multiple Eisenstein series can be written as ordered sums over multitangent functions ( [Bo]), which are for k 1 , . . ., k r ě 2 and τ P H defined by Ψ k 1 ,...,kr pτ q :" ÿ (2.6) These functions were originally introduced by Ecalle and then in detail studied by Bouillot in [Bo].To write G k 1 ,...,kr pτ q in terms of these functions, one splits up the summation in the definition (2.1) into 2 r parts, corresponding to the different cases where either m i " m i`1 or m i ą m i`1 for λ i " m i τ `ni and i " 1, . . ., r (λ r`1 " 0).Then one can check that the multiple Eisenstein series can be written as where the q-series ĝ˚a re given as ordered sums over multitangent functions by ĝ˚p k 1 , . . ., k r q :" ÿ 1ďjďr 0"r 0 ăr 1 ă¨¨¨ăr j´1 ăr j "r m 1 ą¨¨¨ąm j ą0 Further, one can show ([B1, Construction 6.7]) that the q-series ĝ˚s atisfy the harmonic product formula, e.g.ĝ˚p k 1 qĝ ˚pk 2 q " ĝ˚p k 1 , k 2 q `ĝ ˚pk 2 , k 1 q `ĝ ˚pk 1 `k2 q.We will generalize this construction later in terms of generating series (Lemma 6.21) and then use an analogue of the formula (2.7) as the definition for the combinatorial multiple Eisenstein series.To obtain the statement in Theorem 2.1 one then uses the following theorem.
Theorem 2.2.[Bo,Theorem 6] For k 1 , . . ., k r ě 2 with k " k 1 `¨¨¨`k r the multitangent function can be written as Moreover, the terms containing Ψ 1 pτ q vanish.This theorem can be proven by using partial fraction decomposition (see Example 2.3) and then using the shuffle product to show that the coefficient of Ψ 1 pτ q vanishes.Applying Theorem 2.2 to (2.8), we see by (2.5), that the ĝ˚c an be written as a Z-linear combination of ĝ.This proves Theorem 2.1, since one can also show that all the appearing multiple zeta values have the correct depth.

Moulds, bimoulds and quasi-shuffle products
First, we will recall some basic facts on quasi-shuffle products ( [Bo2], [H], [HI]).Let L be a countable set whose elements we will refer to as letters.A monic monomial in the noncommutative polynomial ring QxLy will be called a word and we denote the empty word by 1. Suppose we have a commutative and associative product ˛on the vector space QL.Then the quasi-shuffle product ˚˛on QxLy is defined as the Q-bilinear product, which satisfies 1 ˚˛w " w ˚˛1 " w for any word w P QxLy and aw ˚˛bv " apw ˚˛bvq `bpaw ˚˛vq `pa ˛bqpw ˚˛vq for any letters a, b P L and words w, v P QxLy.This gives a commutative Q-algebra pQxLy, ˚˛q, which is called quasi-shuffle algebra.Moreover, one can equip this algebra with the structure of a Hopf algebra [H,Section 3], where the coproduct is given for w P QxLy by the deconcatenation coproduct ∆pwq " ÿ uv"w u b v .
(3.2) A well-known example is the shuffle Hopf algebra.Define the product on QL by a ˛b " 0 for all a, b P L. Then the corresponding quasi-shuffle product ˚˛on QxLy is the shuffle product, usually denoted by ¡.The antipode in the shuffle Hopf algebra is given by Spa 1 . . .a r q " p´1q r a r . . .a 1 , a 1 , . . ., a r P L, so the defining property of S yields the following relations in QxLy.
Lemma 3.1.For any non-empty word w " a 1 . . .a r in QxLy, it is To work with quasi-shuffle products it is convenient to consider generating series.For this, we will introduce the notion of moulds and bimoulds, which were introduced by Ecalle.We refer to the article [Bo2] for a good overview on mould theory and a thorough list of reference for the original works of Ecalle.
Definition 3.2.Let A be a unital Q-algebra.A family Z " pZ prq q rě0 with Z p0q P A and Z prq P A X 1 , . . ., X r for r ě 1 will be called a mould with values in A.
Given a mould Z " pZ prq q rě0 we will call the Z prq the depth r part of Z.All moulds considered in this article satisfy Z p0q " 1, so we usually just give the depth r ě 1 parts when defining moulds.Since the depth is clear from the number of variables we will just write ZpX 1 , . . ., X r q instead of Z prq pX 1 , . . ., X r q in the following.Let Z " pZ prq q rě0 be an A-valued mould, then we define for r ě 1 and k 1 , . . ., k r ě 1 the elements zpk 1 , . . ., k r q P A as the coefficients of its depth r part ZpX 1 , . . ., X r q ": ÿ (3.3) Consider the set of letters L z " tz k | k ě 1u.Then for any commutative and associative product ˛on QL z we obtain a quasi-shuffle algebra pQxL z y, ˚˛q.
Definition 3.3.Let A be an unital Q-algebra, Z an A-valued mould with coefficients z as defined in (3.3), and ˛a commutative and associative product on QL z .
(i) The mould Z is called ˛-symmetril if the coefficient map ϕ Z : QxL z y Ñ A given on the generators by ϕ Z p1q " 1 and is an algebra homomorphism from pQxL z y, ˚˛q to A. (ii) If ˛is given by z k 1 ˛zk 2 " z k 1 `k2 , then we call a ˛-symmetril mould symmetril.(iii) If the product ˛is given by z k 1 ˛zk 2 " 0, then we call a ˛-symmetril mould symmetral.
Since we usually have Z p0q " 1, the first term (j " 0) in the above sum is simply given by Z 2 pX 1 , . . ., X r q and the last term (j " r) is given by Z 1 pX 1 , . . ., X r q.Equipped with this product the space of all (A-valued) moulds becomes a non-commutative Q-algebra.Definition 3.4.(i) For a mould Z we define the mould Z 7 by Z 7 pX 1 , . . ., X r q " ZpX 1 `¨¨¨`X r , . . ., X 1 `X2 , X 1 q.
(ii) Let F " ř rě0 a r T r P A T be a formal power series with coefficients in A. We can view F as a mould with values in A, which we also denote by F and which is in depth r ě 0 defined by F prq pX 1 , . . ., X r q " a r .We call such a mould a constant mould.Also notice that the product of two constant moulds is exactly the constant mould coming from the product of their power series.(iii) For an A-valued mould Z with coefficients (3.3) we define the constant mould Γ Z by Γ Z :" (3.4) (iv) For an A-valued mould Z define the mould i.e. explicitly we have Moreover, define its coefficients z γ pk 1 , . . ., k r q P A by Conversely, the mould Z can also be written in terms of Z γ ZpX 1 , . . ., X r q " r ÿ j"0 γZ j Z γ pX r , X r´1 ´Xr , . . ., X j`1 ´Xj`2 q , (3.5) where (by using [HI,(32)] for the last equation) we have Definition 3.5.Let A be a unital Q-algebra and Z an A-valued mould.We say that the mould Z satisfies the extended double shuffle equations if the mould Z is symmetril and the mould Z γ is symmetral.
Example 3.6.For a mould Z, the extended double shuffle equations in depth two are (3.7) The motivating example for these equations is the mould of (stuffle regularized) multiple zeta values z, whose depth r part is defined by The mould z satisfies the extended double shuffle equations ( [IKZ], [E]) and the corresponding relations obtained for multiple zeta values are exactly the extended double shuffle relations mentioned in the introduction.Definition 3.7.Let A be a unital Q-algebra.A bimould with values in A is a family B " pB prq q rě0 with B p0q P A and As is the case of moulds, we will call B prq the depth r part of B and since the depth is clear from the number of variables and we will just write B `X1 ,...,Xr Y 1 ,...,Yr ˘instead of B prq `X1 ,...,Xr Y 1 ,...,Yr ȋn the following.Moreover, all appearing bimoulds will also satisfy B p0q " 1, hence we often restrict to the case r ě 1 when defining bimoulds.
For bimoulds, we will consider the alphabet which can be seen as a generalization of L z .For a commutative and associative product ǫn QL bi z we obtain a quasi-shuffle algebra pQxL bi z y, ˚˛q.Definition 3.8.Let A be a unital Q-algebra, B a A-valued bimould, and ˛a commutative and associative product on QL bi z .(i) If the coefficients b `k1 ,...,kr d 1 ,...,dr ˘P A of B in depth r are given by then we define the coefficient map of B as the Q-linear map ϕ B : QxL bi z y Ñ A on the generators by ϕp1q " 1 and , then we call a ˛-symmetril bimould symmetril.(iv) If the product ˛is given by z k 1 d 1 ˛zk 2 d 2 " 0, then we call a ˛-symmetril bimould symmetral.If B is symmetril, then in depth two we have as an analogue of the first equation in (3.7) ´X2 .
Similar as for moulds, we define the product of two bimoulds B and C as the bimould B ˆC given by Proposition 3.9.If B and C are ˛-symmetril (bi)moulds then B ˆC is ˛-symmetril.
Proof.Let ϕ B , ϕ C be the coefficient maps of the (bi)moulds B and C and write m : A b A Ñ A for the multiplication on A. Then we see by definition that the coefficient map of B ˆC is the convolution product of ϕ B and ϕ C , i.e.
where ∆ is the coproduct (3.2) on pQxLy, ˚˛q for L " L z or L " L bi z .This shows that ϕ BˆC : pQxLy, ˚˛q Ñ A is an algebra homomorphism if ϕ B and ϕ C are and therefore B ˆC is ˛-symmetril.
There is another important property of bimoulds, which is closely related to the conjugation of partitions (see [B1], [B3], [BI] and [Bri]).
An explicit formula for the coefficients on the right-hand side of (3.8) can be found in [BI,Remark 3.14], where the swap is denoted by the involution ι and the coefficients b are denoted by P s .

From moulds to bimoulds and the bimould b
Let b be a Q-valued mould, which satisfies the extended double shuffle equations and is in depth one given by bpXq " ˙. (4.1) In particular, the coefficients β of b (as defined in (3.3)) are a Q-valued solution to the extended double shuffle equations.), we will give a short explanation how to obtain such an element.In [R, section IV], the space DM λ pQq Ă QxxL x yy is introduced, where L x " tx 0 , x 1 u.It is then shown that the space DM λ pQq is non-empty, so we can pick an element for λ " βp2q " ´1 24 , i.e. b P DM ´1 24 pQq.There is a canonical projection π z : QxxL x yy Ñ QxxL z yy, which is given on the generators by x k´1 0 x 1 Þ Ñ z k and maps each word ending in x 0 to 0. So applying the map yields a family of generating series bpX 1 , . . ., X r q P QrX 1 , . . ., X r s, which defines a mould b satisfying the extended double shuffle equations.
In [D, §5], the space M µ pQq of associators is defined for each µ P Q.It is shown that the space M µ pQq is non-empty, thus we choose an element b P M 1 pQq.By [F, Cor 0.4], there is an embedding M 1 pQq ãÑ DM pQq (the definition of DM λ pQq in [F] slightly differs from the original one given in [R], thus one has to be careful with the signs).So take the image of b under the embedding and proceed as before to obtain a mould b with values in Q satisfying the extended double shuffle equations.Both approaches do not provide an explicit construction of such a solution, this is an open problem so far.In low depths, there exist explicit rational solutions ( [E], [Br], [GKZ]), which then gives possible candidates for bpX 1 , . . ., X r q in the case r ď 3. See also [B3,Section 3] for an explicit expression of the bimould b in depth two coming from the solution presented in [GKZ] or see [BKM] on how to construct directly such a bimould in depth two out of a power series satisfying the Fay-identity (e.g. a variant of coth).
The mould b is not unique, there are different choices starting in weight 8.In the following, we will fix a mould b with values in Q, which satisfies the extended double shuffle equations and which is given by (4.1) in depth one.In particular, everything we will define in the following will depend on this choice.
The following gives natural constructions to obtain bimoulds out of moulds.
Definition 4.2.(i) For a mould Z we define the two bimoulds X Z and Y Z by (ii) For a mould Z we define the bimould B Z by so explicitly we have where the coefficients γ Z i are given by (3.4).This construction will be used to obtain a bimould version of b in Definition 4.6.We will show that B Z is always a swap invariant bimould and, if Z satisfies the extended double shuffle relations, then B Z is additionally symmetril.
Proposition 4.3.For any mould Z the bimould B Z is swap invariant.
Proof.The swap of B Z is given by Making the change of variables j 1 " r ´j `i, we see that above sum equals Proposition 4.4.If a mould Z satisfies the extended double shuffle equations, then the bimould B Z is symmetril.
Proof.Let Z satisfy the extended double shuffle equations, so Z is a symmetril mould and Z γ is a symmetral mould.We immediately obtain that X Z is a symmetril bimould and Y Zγ is a symmetral bimould.If a bimould does not depend on the variables X i , symmetrility is equivalent to symmetrality.In particular, the bimould Y Zγ is also symmetril.In (4.2) we see that B Z " Y Zγ ˆXZ , so by Proposition 3.9 also B Z is symmetril.
Remark 4.5.In [BIM], the relationship between the classical extended double shuffle equations and the relations of the coefficients of swap invariant and symmetril bimoulds will be explained in detail.In particular, it will be shown that in the special case of moulds our Definition 3.5 coincides with the classical notion of extended double shuffle equations used in [IKZ] and [R].
Definition 4.6.For the fixed Q-valued mould b we define its corresponding bimould by b " B b .By abuse of notation we denote the mould and the bimould by b, it will become clear from the the set of variables which one is meant.Explicitly, we have where γ k " γ b k with the notation in (3.4), i.e. with (4.1) we have Corollary 4.7.The bimould b is swap invariant and symmetril.
Proof.This is just a special case of Proposition 4.3 and 4.4.

The bimould g
For m ě 1, we define the following power series in Q q X, Y L m ˆX Y ˙" e X`mY q m 1 ´eX q m , (5.1) which will be used in the construction of combinatorial multiple Eisenstein series and which is the building block of the following bimould.
The coefficients g of g as defined in Definition 3.8 (i) are explicitly given by m dr r pk r ´1q! q m 1 n 1 `¨¨¨`mrnr . (5.2) These coefficients are generalizations of the q-series defined in (2.3).The coefficient of q n is given by the sum over all m 1 n 1 `¨¨¨`m r n r " n with m 1 ą ¨¨¨ą m r ą 0, n 1 , . . ., n r ą 0, i.e. all partitions of n with r different parts m 1 , . . ., m r and multiplicities n 1 , . . ., n r .This sum is invariant under the conjugation of partitions, which on the level of the generating series g corresponds exactly to the swap invariance (3.8).This describes a combinatorial proof of Proposition 5.2.Moreover, see [B3] for the interpretation of the coefficients of the bimould g as a generalization of the generating series of classical divisor-sums and their derivatives.
The bimould g is not symmetril, but we can define a product ˆsuch that it becomes ˆsymmetril.For this, we need the following property of the series L m defined in (5.1).
Lemma 5.3.For all m ě 1 we have where bpXq " ´řkě2 2k! X k´1 is the depth one part of the mould b defined in (4.1).

Proof. By direct calculation one checks that
which gives the above formula by using and the parity of b.From this lemma, one can deduce the quasi-shuffle product satisfied by the coefficients g of g.Explicitly, define for k 1 , k 2 , j ě 1 the rational numbers and define the commutative and associative product ˆon QL bi z by (5.4) Proposition 5.4.The bimould g is ˆ-symmetril.
Proof.This follows from [B1, Theorem 3.6] and is a consequence of Lemma 5.3.For example, in lowest depth we have Considering the coefficients of (5.5) one sees that ˆin (5.4) gives The general case can be proven by induction over the depth.Proposition 5.4 shows the relationship between the bimould g and the depth one part of the mould b.This will play a crucial role in the construction of the combinatorial multiple Eisenstein series.

Combinatorial multiple Eisenstein series
In this section, we will introduce combinatorial (bi-)multiple Eisenstein series, which are the coefficients of the bimould G. Before we can give the definition of G we need to introduce three other bimoulds b, L m , and g ˚, which all depend on a fixed choice of a symmetril and swap-invariant bimould b given in Definition 4.6.
6.1.The bimoulds b, L m , and g ˚.Similar as in Definition 3.4 (ii) we can view the power series exp `´T 2 ˘P Q T as a constant bimould.Moreover, define for any mould Z the mould Z ´for each r ě 1 by Z ´pX 1 , . . ., X r q " Zp´X 1 , . . ., ´Xr q.
With this we define the following analogue of b " B b " Y bγ ˆXb .
i.e. for each r ě 1 we have bˆX For m ě 1, let L m be the bimould given in depth r ě 1 by Observe that the depth one part of L m is exactly given by the series L m defined in (5.1).
Remark 6.2.(i) The bimould L m can be also defined by using the flexion markers introduced in [E] (cf.[Bo,Section 7.5.3]).
(ii) The definition of L m is inspired by the calculation of the Fourier expansion of multiple Eisenstein series.First observe that the power series L m can be seen as the generating series of the monotangent functions for Y " 0. Namely, define the 'combinatorial version' of the monotangent function Ψ comb k pτ q " 1 pk´1q!ř dą0 d k´1 q d for k ě 1 by using simply the right hand side of the Lipschitz formula (2.2).Then we see that So in analogy to Theorem 2.2, the L m can be seen as the generating series of the combinatorial version of the multitangent functions.In particular, the trifactorization of the mould of monotangent functions (used to prove Theorem 2.2) in [Bo,Theorem 5 & 6 ] is similar to our definition of L m .The mould consisting of multiple zeta values in [Bo] is in the definition of L m replaced by the mould b.Moreover, we omit the constant term for Ψ 1 pτ q, this is necessary for our construction of combinatorial multiple Eisenstein series for arbitrary indices (see the discussion before Remark 6.14 in [B1]).
Definition 6.3.We define the bimould g ˚in depth r ě 1 by The definition of the bimould g ˚is inspired by the definition of the classical g ˚in (2.8).In particular, we will show that the bimould g ˚is symmetril (Proposition 6.22).6.2.The bimould G and combinatorial (bi-)multiple Eisenstein series.In analogy to (2.7) we define the bimould G as the product of g ˚and b.
Definition 6.4.(i) We define the bimould G by G " g ˚ˆb .

˙.
The mould given by their generating series will also denoted by G, i.e.

˙.
The main result of this work is the following.
Proof.We will show that g ˚is symmetril (Proposition 6.22).Since b is also symmetril, we deduce by Proposition 3.9 that G " g ˚ˆb is symmetril.For swap invariance we will show that G can be written as a sum of swap invariant bimoulds G j (Proposition 6.27, Theorem 6.26) and therefore G is itself swap invariant.
Before presenting the necessary results for the proof of Theorem 6.5, we will give some examples and consequences.
Example 6.6.(i) In depth r " 1 we have g So the coefficients are for k ě 1, d ě 0 given by ˘, which can also be obtained from (6.2). (ii) In depth r " 2 the bimould G is given by This gives an explicit expression of G `k1 ,k 2 d 1 ,d 2 ˘in terms of the β and the q-series g, which we will omit here.(iii) In depth r " 2 the swap invariance reads In Proposition 6.7 this formula is used to give an analogue of the shuffle product formula for combinatorial multiple Eisenstein series.(iv) We saw in Example 2.3 that G 3,2 is given by G 3,2 pτ q " ζp3, 2q `3ζp3qĝp2q `2ζp2qĝp3q `ĝp3, 2q .
Notice that βp3q " 0 and therefore this is exactly the same expression after replacing ζ by the rational numbers β and ĝ by g.
As an analogue of the double shuffle equations of multiple zeta values in depth two (1.3), the combinatorial bi-multiple Eisenstein series satisfy the following.
Proof.The first equality is just a direct consequence of the symmetrility.To show the second equality, first use the swap invariance to get G `k1 ´1˘a nd then evaluate this product by the first equality.Using then again the swap invariance in depth one and depth two (6.3) yields the result.Remark 6.8.Proposition 6.7 shows that the combinatorial bi-multiple Eisenstein series in depth ď 2 give a realization of the formal double Eisenstein space introduced in [BKM].This space is exactly defined by formal symbols satisfying the relation in Proposition 6.7.
One can obtain an analogue for the double shuffle relations in arbitrary depths with the same argument as in the proof of Proposition 6.7.For example, the equation (1.9) is obtained in this way.
Example 6.9.Evaluating Gp2qGp2, 1q in the two different ways described above and writing out the Fourier expansion yields: From this equation we can obtain relations among the coefficients β in lower weight without using their explicit expression.We get βp2q " ´1 24 , βp1, 1q " 1 48 and 2βp1, 3q `3βp2, 2q 6βp3, 1q " 1 1152 " 1 2 βp2q 2 .It might be interesting to understand in general, which relations among the β can be obtained from the relations among the G.
Definition 6.10.The Q-vector space spanned by all combinatorial bi-multiple Eisenstein series is defined by and the homogeneous subspace of weight k ě 0 is given by G bi 0 " Q and for k ě 1 by The subspace spanned by all combinatorial multiple Eisenstein series will be denoted by k .Remark 6.11.We expect that all relations among the combinatorial multiple Eisenstein series come from the swap invariance and symmetrility.In particular, this would imply that the combinatorial bi-multiple Eisenstein series are graded by weight, i.e. we expect G bi ?" À kě0 G bi k .Proposition 6.12.Both G bi and G are Q-algebras containing the algebra of (quasi-) modular forms with rational coefficients, given by QrGp2q, Gp4q, Gp6qs.
Proof.This follows immediately from the symmetrility of G and (6.2).It can be also obtained from Proposition 6.15 (i) below.Proposition 6.13.For k ě 1, d ě 0 with k `d even, G `k,...,k d,...,d ˘is a quasi-modular form.Proof.By a classical result for quasi-shuffle algebras ( [HI,(32)]),the generating series of G `k,...,k d,...,d ˘can be written as (6.4) By (6.2) the G `rk rd ˘are quasi-modular for even k `d and therefore by (6.4) G `k,...,k d,...,d ˘are also quasi-modular forms of weight rpk `dq.
Example 6.14.We give one explicit example for k " d " 1 and n " 2: Here the first equality comes from the definition of G `1,1 ˘, the second equality follows from using explicit values for β ,which are unique up to weight 7, and the swap invariance of g.The third equality comes from 6.4, but could also be obtained from Proposition 5.4.For the last equation we used the swap invariance of G and (6.2).
For some indices one can also give an explicit formula for the G in terms of the q-series g, e.g. in the case k " 2, d " 0 one can show that To obtain this formula one shows that the generating series over all r ě 1 of the coefficients of X 1 . . .X r in L m `X1 ,...,Xr 0,...,0 ˘has a product expression in terms of the L m .Using the Weierstrass product expression of sin together with our construction then yields the claim after some calculations.It would be interesting to know if in general there are similar expressions for Gp2k, . . ., 2kq in analogy to the explicit evaluations of ζp2k, . . ., 2kq for k ě 1.
By construction the combinatorial bi-multiple Eisenstein series G can be written as rational linear combinations of the q-series g defined in (5.2).The following Proposition shows that also the converse is true.Proposition 6.15.For all k 1 , . . ., k r ě 1 and d 1 , . . ., d r ě 0 we have In particular, the combinatorial bi-multiple Eisenstein series span the space Z q of q-analogues of multiple zeta values defined in [BK2].
Proof.In depth one we have by definition G `k d ˘" g `k d ˘`µ for some µ P Q (see (6.2)), thus it is g `k d ˘P G bi for all k ě 1, d ě 0.Moreover, we obtain immediately from the construction of G that for all k 1 , . . ., k r ě 1 and d 1 , . . ., d r ě 0 . ., k r d 1 , . . ., d r ˙`pterms only involving g of smaller depths and weightsq.
So induction on the depth shows that each q-series g is a Q-linear combination of combinatorial bi-multiple Eisenstein series.The last statement follows from [BK2, Theorem 1], where it is shown that the q-series g span the space Z q , i.e. we get G bi " Z q .
Remark 6.16.(i) The similar argument as in Proposition 6.15 also shows gpk 1 , . . ., k r q P G for all k 1 , . . ., k r ě 1.Also the converse holds, i.e. every combinatorial multiple Eisenstein series is also a Q-linear combination of the single indexed g (this follows from equation (6.15)).This is in contrast to the shuffle ( [BT]) and stuffle ( [B1]) regularized multiple Eisenstein series, where double indexed g are needed when k j " 1 for some j.In particular, we have G " Z q where Z q is defined in [BK2].As a consequence of this, one would expect G ? " G bi .This was first conjectured in [B1] (see also [BK2,Conjecture 5]).(ii) As explained in Remark 6.11, we expect that G bi is graded by weight.By Proposition 6.15, the dimensions of the homogeneous spaces G bi k should coincide with the conjectured dimensions of the weight-graded parts of Z q given in [BK2, Conjecture 3] (and similarly for the associated depth graded parts).
We will explain now why the combinatorial multiple Eisenstein series interpolate between the rational solution β and multiple zeta values.In the case k 1 ě 2, k 2 , . . ., k r ě 1 we get as a direct consequence of the proof of Proposition 6.15 and (2.4) that lim qÑ1 p1 ´qq k 1 `¨¨¨`kr Gpk 1 , . . ., k r q " ζpk 1 , . . ., k r q . (6.5) The limit is independent of the choice of the rational solution to the double shuffle equations b, since the limit q Ñ 1 considers just the highest depth term of the q-series g in G.In the case k 1 " 1 this limit does not exist, but we can consider a regularized limit, which we will describe now.Using the notation as in Section 3 we can, as in the introduction, view the combinatorial multiple Eisenstein series as a Q-linear map defined on the generators by Since G is symmetril, G gives an algebra homomorphism from pH 1 , ˚q to G. Due to H 1 " H 0 rz 1 s (cf.[IKZ, Proposition 1]) we can write w " z k 1 . . .z kr P H 1 for any k 1 , . . ., k r ě 1 uniquely as w " ř r j"0 w j ˚z˚j 1 with w j P H 0 .Then we define the regularized version of the limit (6.5) as lim qÑ1 ˚p1 ´qq k 1 `¨¨¨`kr Gpk 1 , . . ., k r q :" lim qÑ1 p1 ´qq k 1 `¨¨¨`kr Gpw 0 q " ζpw 0 q . (6.7) Notice that if k 1 ě 2, then w " w 0 and thus (6.7) is equal to (6.5).
Proof.First notice that the constant terms of the combinatorial multiple Eisenstein series are by construction the coefficients in b `X1 ,...,Xr 0,...,0 ˘.The bimould b was defined by the mould b (Definition 4.6), which satisfies the extended double shuffle equations.Since the mould b γ is symmetral and b γ p0q " 0 (as βp1q " 0), we inductively get b γ p0, . . ., 0q " 0. We deduce b `X1 ,...,Xr 0,...,0 ˘" bpX 1 , . . ., X r q which shows the first equation.For the second equation observe that the stuffle regularized multiple zeta values ζ ˚are essentially defined in the same way as we constructed the regularized limit (6.7).Then since G and ζ ˚are algebra homomorphisms, we obtain the claim from (6.5).
Proof.If f pτ q " a 0 `řně1 a n q n is modular of weight k then f p´1 τ q " τ k f pτ q, i.e.
The statement then follows from Proposition 6.17.
6.3.Symmetrility of L m , g ˚and G.In this subsection, we will give the proofs for the symmetrility of previously mentioned bimoulds.
Proof.By replacing q m " e ´T in L m `X 0 ˘" e X q m 1´e X q m , we obtain a new series For each r ě 1, we define a multiple bi-version of L T as ˙LT pX j q bˆX r ´Xj , . . ., X j`1 ´Xj ´Yr , . . ., ´Yj`1 ˙.
Then after the change of variables q m " e ´T and multiplication with exppmpY 1 `¨¨¨`Y r qq, we obtain precisely the bimould L m .Moreover, let b T , bT and M T be the bimoulds given in depth r ě 1 by We will show that the bimould L T has the following product representation Since all bimoulds on the right hand side of the equation are symmetril, by Proposition 3.9 also L T is a symmetril bimould.Substituting back e ´T " q m and multiplying by exppmpY 1 Ỳr qq gives the symmetrility of the bimould L m .In depth one, we compute (6.9) Substituting (6.9) in the left hand side of (6.8), we have to show in some given depth r ě 1 r ÿ Then apply symmetrility to the terms bpX a ´Xj , . . ., X j´1 ´Xj q and ´bpT ´Xj q for a P t1, . . ., ju in the first row and to the terms ´bpT ´Xj q and bpX r´b ´Xj , . . ., X j`1 ´Xj q for b P t0, . . ., r ´ju in the second row (after making use of the identity bpXq " ´bp´Xq).
Finally, rewrite the equation in terms of the mould b γ by using (3.5).Since the mould b γ is symmetral, it satisfies by Lemma 3.1 r´b ÿ j"a´1 p´1q j b γ pX j , X j´1 , . . ., X a qb γ pX j`1 , X j`2 , . . ., X r´b q " 0 (6.11) for all 1 ď a ď r ´b ď r.Frequently applying the relation (6.11) proves the above equation except for the terms, where no mould b depending on some of the variables X 1 , . . ., X r appears.To show that these terms also vanish, we use an explicit expression for the generating series of γk " γb k defined in (3.6) γpXq " The following expression of the Gamma function (c.f.[IKZ,(2.1)]) ţogether with the equality βpnq " ζpnq p2πiq n for n even, gives Using the definition of b as a product of three moulds given in (6.1) one can write the remaining terms in (6.10) as products of moulds for which one can show that they cancel out by using the explicit formula (6.12) together with some straightforward calculation.
Lemma 6.21.Let B m be a family of bimoulds which are ˛-symmetril for all m ě 1.Then the bimould C M defined by ..,Xr Y 1 ,...,Yr ˘" 0 for r ě 1 and C p0q 1 " 1, thus C 1 is a ˛-symmetril bimould.Moreover, one obtains from direct calculations C M `1 " B M ˆCM .Therefore, induction on M and Proposition 3.9 yields the ˛-symmetrility of the bimould C M for all M ě 1. Proposition 6.22.The bimould g ˚is symmetril.
Proof.Choosing B m " L m in Lemma 6.21 and taking the limit M Ñ 8 gives the bimould g ˚.By Lemma 6.20 the bimoulds L m are symmetril for all m ě 1, thus we obtain that g ˚is symmetril.
Remark 6.23.The bimould g ˚can be seen as variant of the bimould g which is symmetril instead of ˜-symmetril.It should be remarked that this correction is a completely different to the one obtained by using the maps log and exp from [H] and [HI], which enables one to switch between different quasi-shuffle products over the same alphabet.This other approach is illustrated in [B1,Remark 6.6 Definition 6.25.For j ě 0 we define the bimould G j " pG prq j q rě0 as follows.In the case j " 0 we set G 0 " b and in general G prq j " 0 for j ą r.If 1 ď j ď r we define Notice that we have G prq r " g prq for any r ě 1, i.e. the G j can be seen as an interpolation between the swap invariant bimoulds b and g.Using the swap invariance of b and g we get the following more general result.
Theorem 6.26.The bimould G j is swap invariant for any j ě 0.
Proof.Since G 0 " b we can assume 1 ď j ď r in the following.For 1 ď a ď b ď r we will use the notation X b a :" X a ´Xb and Y b a :" Y a `¨¨¨`Y b .The L m i can then be written as By the definition of the bimould g in 5.1 as a sum over the L m , we therefore obtain We want to check that G j satisfies (3.8), i.e. that it is invariant under the simultaneous change of variables X j Ñ Y 1 `¨¨¨`Y r´j`1 " Y r´j`1 1 and Y j Ñ X r´j`1 ´Xr´j`2 " X r´j`2 r´j`1 for j " 1, . . ., r (here X r`1 :" 0), which imply X b a Ñ Y In the last equality we used that (by Definition 4.6) both b and b, and therefore C r 1 ,...,r j n 1 ,...,n j , vanish under any B BX i B BY i and that the g terms are independent of X i if i R tn 1 , . . ., n j u, so they vanish under B BX i B BY i in these cases.Since G is the sum of the G j (Proposition 6.27), we obtain the formula (6.17).Proposition 6.30.For k 1 , . . ., k r ě 1 we have q d dq Gpk 1 , . . ., k r q " Gp2qGpk 1 , . . ., k r q ´ÿ 1ďjďr a`b"k j `2pa ´1qGpk 1 , . . ., k j´1 , a, b, k j`1 , . . ., k r q ´ÿ 1ďiăjďr a`b"k j `1 k i Gpk 1 , . . ., k i `1, . . ., k j´1 , a, b, k j`1 , . . ., k r q ´ÿ 1ďiďr k i Gpk 1 , . . ., k i `1, . . ., k r , 1q ´Gpk 1 , . . ., k r , 2q . (6.18) In particular, the space G is closed under q d dq .Proof.Since G is symmetril, we have
With the interpretation of G as an algebra homomorphism from pH 1 , ˚q to G in (6.6), we can give the following reinterpretation and consequence of Proposition 6.30.Corollary 6.31.(i) For w P H 1 the derivative of Gpwq is given by q d dq Gpwq " Gpz 2 ˚w ´z2 ¡ wq .
(ii) Let h : H 1 Ñ H 1 be the linear map defined on the generators by h : w Þ ÝÑ z 2 ˚w ´z2 ¡ w .
Then for any v, w P H 1 we have hpw ˚vq ´hpwq ˚v ´w ˚hpvq P ker G .
Proof.The equation in (i) is a direct consequence of 6.30 since the sums on the right-hand side of (6.18) correspond exactly to those indices which appear in the shuffle product of z 2 " xy with z k 1 . . .z kr " x k 1 ´1y . . .x kr´1 y.For (ii) we use that G is an algebra homomorphism and q d dq is a derivation on Q q .By (i) we have q d dq Gpwq " Gphpwqq and therefore get Gphpw ˚vq ´hpwq ˚v ´w ˚hpvqq " 0.
Notice that the map h is not a derivation on pH 1 , ˚q, i.e. the relations we obtain among the combinatorial multiple Eisenstein series from Corollary 6.31 (ii) are non-trivial.For example, for v " w " z 1 we get Gp4q " 2Gp2, 2q ´2Gp3, 1q.This is the first relation, and the only in weight 4, among combinatorial multiple Eisenstein series, since the q-series gpk 1 , . . ., k r q do not satisfy relations in lower weight (see [BK,(1.9)]).It would be interesting to see if one can describe Gpv ˚w ´v ¡ wq for arbitrary w, v P H 1 explicitly and if this can be used to obtain relations among combinatorial multiple Eisenstein series similar to Corollary 6.31 (ii).Remark 6.32.In [F2] the Alekseev-Torossian associator, whose coefficients satisfy the extended double shuffle equations, is computed.It turns out that in depth 1, it satisfies also the additional conditions given in (4.1) (compare to [F2,Example 4.1]).In general, the coefficients of the AT associator are not rational.But replacing the rational solution b to the extended double shuffle equations by the AT associator gives another family of q-series whose generating series are symmetril and swap invariant.