Related papers: Symmetric multiple Eisenstein series
In their seminal paper "Double zeta values and modular forms" Gangl, Kaneko and Zagier defined a double Eisenstein series and used it to study the relations between double zeta values. One of their key ideas is to study the formal double…
We study the double shuffle relations satisfied by the double zeta values of level 2, and introduce the double Eisenstein series of level 2 which satisfy the double shuffle relations. We connect the double Eisenstein series to modular forms…
We study a certain class of q-analogues of multiple zeta values, which appear in the Fourier expansion of multiple Eisenstein series. Studying their algebraic structure and their derivatives we propose conjectured explicit formulas for the…
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 $\mathbb{Q}$-valued solution of the extended double shuffle equations. These…
In [CCHT25], the authors introduced multiple Eisenstein series of arbitrary rank in positive characteristic and the $q$-shuffle algebra $\mathcal{E}$ associated with them. In the present paper, we establish a class of linear independence…
We introduce the algebra of formal multiple Eisenstein series and study its derivations. This algebra is motivated by the classical multiple Eisenstein series, introduced by Gangl-Kaneko-Zagier as a hybrid of classical Eisenstein series and…
Multiple Eisenstein series are holomorphic functions in the complex upper-half plane, which can be seen as a crossbreed between multiple zeta values and classical Eisenstein series. They were originally defined by Gangl-Kaneko-Zagier in…
Bachmann and Tasaka discovered a relationship between multiple Eisenstein series (MES) of level 1 and formal iterated integrals corresponding to multiple zeta value. They also constructed shuffle regularized MES of level 1, which satisfies…
From the theory of modular forms, there are exactly $[(k-2)/6]$ linear relations among the Eisenstein series $E_k$ and its products $E_{2i}E_{k-2i}\ (2\le i \le [k/4])$. We present explicit formulas among these modular forms based on the…
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…
In this paper, we study multiple Eisenstein series, which build a natural bridge between the theory of multiple zeta values and modular forms. We prove a large family of relations among these series and propose an explicit conjectural…
We introduce the balanced multiple q-zeta values. They give a new model for multiple q-zeta values, whose product formula combines the shuffle and stuffle product for multiple zeta values in a natural way. Moreover, the balanced multiple…
This article proposes a new approach to studying the spectral Eisenstein series of weight $k$ on a congruence subgroup of $\text{SL}_2(\mathbb{Z})$ using Hecke's theory of Eisenstein series for the principal congruence subgroups. Our method…
We study the multiple Eisenstein series introduced by Gangl, Kaneko and Zagier. We give a proof of (restricted) finite double shuffle relations for multiple Eisenstein series by developing an explicit connection between the Fourier…
The purpose of this article is to give a simple and explicit construction of mock modular forms whose shadows are Eisenstein series of arbitrary integral weight, level, and character. As application, we construct forms whose shadows are…
We show that every elliptic modular form of integral weight greater than $1$ can be expressed as linear combinations of products of at most two cusp expansions of Eisenstein series. This removes the obstruction of nonvanishing central…
We introduce an algebra which describes the multiplication structure of a family of q-series containing a q-analogue of multiple zeta values. The double shuffle relations are formulated in our framework. They contain a q-analogue of…
We study the algebra of certain $q$-series, called bi-brackets, whose coefficients are given by weighted sums over partitions. These series incorporate the theory of modular forms for the full modular group as well as the theory of multiple…
The aim of this paper is to study certain multiple series which can be regarded as multiple analogues of Eisenstein series. As a prior research, the second-named author considered double analogues of Eisenstein series and expressed them as…
We introduce the formal double Eisenstein space $\mathcal{E}_k$, which is a generalization of the formal double zeta space $\mathcal{D}_k$ of Gangl-Kaneko-Zagier, and prove analogues of the sum formula and parity result for formal double…