Related papers: Combinatorial multiple Eisenstein series
In this paper we present a method for constructing multiple-sum $q$-series for what is known as Mixed Mock Modular forms. We also present some multi-sum analogues of the Durfee identity, and discuss a construction of its combinatorial…
The low-energy expansion of closed-string scattering amplitudes at genus one introduces infinite families of non-holomorphic modular forms called modular graph forms. Their differential and number-theoretic properties motivated Brown's…
We give explicit expressions for MacMahon's generalized sums-of-divisors $q$-series $A_r$ and $C_r$ by relating them to (odd) multiple Eisenstein series. Recently, these sums-of-divisors have been studied in the context of quasimodular…
MacMahon's partition functions and their extensions provide equations that identify prime numbers as solutions. These results depend on the theory of (mixed weight) quasimodular forms on $SL_2(\mathbb{Z})$. Two of the authors, along with…
We study the algebra $\mathcal{I}^{QM}$ of iterated integrals of quasimodular forms for $\operatorname{SL}_2(\mathbb{Z})$, which is the smallest extension of the algebra $QM_{\ast}$ of quasimodular forms, which is closed under integration.…
We present the $\tau$-invariant balanced quasi-shuffle algebra $\mathcal{G}^{\operatorname{f}}$, whose elements formalize (combinatorial) multiple Eisenstein series as well as multiple q-zeta values. In particular,…
A systematic procedure for generating cubic multisections of Eisenstein series is given. The relevant series are determined from Fourier expansions for Eisenstein series by restricting the congruence class of the summation index modulo…
The Newton series which interpolate finite multiple harmonic sums are useful in the study of multiple zeta values (MZV's). In this paper, we prove that these Newton series can be written as multiple series. As an application, we give a…
In this paper we study some combinatorial properties of the shuffle algebra $S_{q_{1},q_{2}}$ with two complex parameters and, in particular, obtain the Hilbert series for $S_{q_{1},q_{2}}/I$, where $I$ is the ideal generated by $z-\lambda$…
Interpolated multiple zeta values can be regarded as interpolation polynomials of multiple zeta values and multiple zeta-star values. In this paper, we give some algebraic relations of interpolated multiple zeta values, such as the…
Following Bachmann's recent work on bi-brackets and multiple Eisenstein series, Zudilin introduced the notion of multiple q-zeta brackets, which provides a q-analog of multiple zeta values possessing both shuffle as well as quasi-shuffle…
In this paper, the extended double shuffle relations for interpolated multiple zeta values are established. As an application, Hoffman's relations for interpolated multiple zeta values are proved. Furthermore, a generating function for sums…
The Fourier coefficient of a second order Eisenstein series is described as a shifted convolution sum. This description is used to obtain the spectral decomposition of and estimates for the shifted convolution sum.
We prove the combinatorial invariance of the coefficient of $q$ in Kazhdan--Lusztig polynomials for arbitrary Coxeter groups. As a result, we obtain the Combinatorial Invariance Conjecture, of Lusztig and of Dyer, also for Bruhat intervals…
A combinatorial study of multiple $q$-integrals is developed. This includes a $q$-volume of a convex polytope, which depends upon the order of $q$-integration. A multiple $q$-integral over an order polytope of a poset is interpreted as a…
Using the concept of mixable shuffles, we formulate explicitly the quantum quasi-shuffle product, as well as the subalgebra generated by primitive elements of the quantum quasi-shuffle bialgebra. We construct a braided coalgebra structure…
We study second-order modular differential equations whose solutions transform equivariantly under the modular group. In the reducible case, we construct all such solutions using an explicit ansatz involving Eisenstein series and the…
In this paper we investigate a result of Ueno on the modularity of generating series associated to the zeta functions of binary Hermitian forms previously studied by Elstrodt et al. We improve his result by showing that the generating…
For the class of quantum integrable models generated from the $q-$Onsager algebra, a basis of bispectral multivariable $q-$orthogonal polynomials is exhibited. In a first part, it is shown that the multivariable Askey-Wilson polynomials…
This paper is a continuation of earlier work of Arslan \cite{Ars}, who introduced the Mahonian number of type $B$ by using a new statistic on the hyperoctahedral group $B_{n}$, in response to questions he suggested in his paper entitled…