Related papers: Eisenstein series associated with $\Gamma_0(2)$

Let $G$ be an affine or hyperbolic rank 2 Kac--Moody group over a finite field $\mathbb F_q$. Let $X=X_{q+1}$ be the Tits building of $G$, the $(q+1)$--homogeneous tree, and let $\Gamma$ be a non-uniform lattice in $G$. When $\Gamma$ is a…

Eisenstein polynomials, which were defined by the second author, are analogues of the concept of an Eisenstein series. The second author conjectured that there exist some analogous properties between Eisenstein series and Eisenstein…

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…

For every positive integral level $k$ we study arithmetic properties of certain holomorphic modular forms associated to modular invariant spaces spanned by graded dimensions of $L_{\hat{sl_2}}(k \Lambda_0)$-modules. We found a necessary and…

We define and study the space of mixed modular symbols for a given finite index subgroup $\Gamma$ of $SL_2(\mathbf{Z})$. This is an extension of the usual space of modular symbols, which in some cases carries more information about…

In this paper we study the modular differential equation $y''+s\,E_4\, y=0$ where $E_4$ is the weight 4 Eisenstein series and $s=\pi^2r^2$ with $r=n/m$ being a rational number in reduced form such that $m\geq 7$. This study is carried out…

Let $\Gamma\subset\mathrm{PSL}_{2}(\mathbb{R})$ be a Fuchsian subgroup of the first kind acting by fractional linear transformations on the upper half-plane $\mathbb{H}$, and let $M=\Gamma\backslash\mathbb{H}$ be the associated finite…

In this paper, we investigate applications of the ordinary derivative operator, instead of the $q$-derivative operator, to the theory of $q$-series. As main results, many new summation and transformation formulas are established which are…

In this paper we introduce two general identities relating Eisenstein series on split classical groups, as well as double covers of symplectic groups. The first identity can be viewed as an extension of the doubling construction introduced…

We introduce the degenerate Bernoulli numbers of the second kind as a degenerate version of the Bernoulli numbers of the second kind. We derive a family of nonlinear differential equations satisfied by a function closely related to the…

For any rational prime $p$, we define a certain $p$-stabilization of holomorphic Siegel Eisenstein series for the symplectic group ${\rm Sp}(2n)_{/\mathbb{Q}}$ of an arbitrary genus $n \ge 1$. In addition, we derive an explicit formula for…

In [B-G1] and [B-G2], Borisov and Gunnells constructed for each level (N > 1) and for each weight (k > 1) a modular symbol with values in $Sk(\Gamma_1(N))$ using products of Eisenstein series. In this paper we generalize this result by…

Let $E(z,s)$ be the non-holomorphic Eisenstein series for the modular group $SL(2,{\mathbb Z})$. The classical Kronecker limit formula shows that the second term in the Laurent expansion at $s=1$ of $E(z,s)$ is essentially the logarithm of…

We use mass formulas to construct minimal parabolic Eisenstein congruences for algebraic modular forms on reductive groups compact at infinity, and study when these yield congruences between cusp forms and Eisenstein series on the…

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…

Modular and quasimodular forms have played an important role in gravity and string theory. Eisenstein series have appeared systematically in the determination of spectrums and partition functions, in the description of non-perturbative…

The Gauge/Bethe correspondence relates Omega-deformed N=2 supersymmetric gauge theories to some quantum integrable models, in simple cases the integrable models can be treated as solvable quantum mechanics models. For SU(2) gauge theory…

We explicitly write down the Eisenstein elements inside the space of modular symbols for Eisenstein series with integer coefficients for the congruence subgroups $\Gamma_0(N)$ with $N$ odd square-free. We also compute the winding elements…

Let $\Gamma$ be a geometrically finite Fuchsian group and suppose that $\chi\colon\Gamma\to\mathrm{GL}(V)$ is a finite-dimensional representation with non-expanding cusp monodromy. We show that the parabolic Eisenstein series for $\Gamma$…

Applying the theory of modular forms and Lambert series manipulations we establish an Eisenstein series identity. From this formula we confirm a Lambert series identity conjectured by Gosper. Another Lambert series identity of Gosper is…