Related papers: Eisenstein series associated with $\Gamma_0(2)$
The existence of periodic solutions in $\Gamma$-symmetric Newtonian systems $\ddot{x}=-\nabla f(x)$ can be effectively studied by means of the $(\Gamma\times O(2))$-equivariant gradient degree with values in the Euler ring $U(\Gamma\times…
We consider a set of generators for the space of Eisenstein series of even weight $k$ for any congruence group $\Gamma$ and study the set of all of their zeros taken for $\Gamma(1)$-conjugates of $\Gamma$ in the standard fundamental domain…
In this note we envisage the relation existing between the Lie Groups and the Theory of Complex Variables. In particular, it is shown that the dimensions of the irreducibles representations of $SU(N)$ may be written in terms of the…
We construct a class of companion elliptic functions associated with the even Dirichlet characters. Using the well-known properties of the classical Weierstrass elliptic function $\wp(z|\tau)$ as the blueprint, we will derive their…
We compute explicit formulae for the constant terms and Fourier coefficients for Eisenstein series on $\operatorname{Sp}(4,\mathbb{R})$, in terms of zeta functions and Whittaker functions. We also develop a generalisation of Ramanujan sums…
We present results for infinite series appearing in Feynman diagram calculations, many of which are similar to the Euler series. These include both one-, two- and three-dimensional series. All these series can be expressed in terms of…
We exhibit a relationship between the massless $a_2^{(2)}$ integrable quantum field theory and a certain third-order ordinary differential equation, thereby extending a recent result connecting the massless sine-Gordon model to the…
We generalize the work of Ohta on the congruence modules attached to elliptic Eisenstein series to the setting of Hilbert modular forms. Our work involves three parts. In the first part, we construct Eisenstein series adelically and compute…
We study the Witten--Reshetikhin--Turaev SU(2) invariant for the Seifert manifold $S^3/\Gamma$ where $\Gamma$ is a finite subgroup of SU(2). We show that the WRT invariants can be written in terms of the Eichler integral of the modular…
Chapter $7$ of Langlands' monograph "On the functional equations satisfied by Eisenstein series" employs a sophisticated residue scheme to construct a portion of the discrete automorphic spectrum. We show, by examples, applications, and…
We find nice representatives for the 0-dimensional cusps of the degree $n$ Siegel upper half-space under the action of $\Gamma_0(\stufe)$. To each of these we attach a Siegel Eisenstein series, and then we make explicit a result of Siegel,…
We show that all Eichler integrals, and more generally all "generalized second order modular forms" can be expressed as linear combinations of corresponding generalized second order Eisenstein series with coefficients in classical modular…
We discuss parabolic versions of Euler's identity e^{it}=cos t + i sin t. A purely algebraic approach based on dual numbers is known to produce a very trivial relation e^{pt} = 1+pt. Therefore we use a geometric setup of parabolic rotations…
We present several identities with a form of polynomials or rational functions that involve Pochhammer and q-Pochhammer symbols and q-binomials (i.e. Gauss polynomials). All these identities were obtained by some analytical methods based on…
In the present paper combinatorial identities involving q-dual sequences or polynomials with coefficients q-dual sequences are derived. Further, combinatorial identities for q-binomial coefficients(Gaussian coefficients), q-Stirling numbers…
Two linear recurrences exhibit mirror symmetry connecting the constants $e$ and $\pi$. When parametrized, their asymptotic connection constants extend to meromorphic functions satisfying additive functional equations with rational…
The aim of the paper is to relate computational and arithmetic questions about Euler's constant $\gamma$ with properties of the values of the $q$-logarithm function, with natural choice of $q$. By these means, we generalize a classical…
The classical Kronecker limit formula gives the constant term of the non-holomorphic Eisenstein series E(z,s) for SL(2,Z) at s=1 in terms of the Dedekind eta function. Here we compute the analagous formula for an Eisenstein series twisted…
We derive weighted summation identities involving the second order recurrence sequence $\{w_n\} =\{ w_n(a,b; p, q)\}$ defined by $w_0 = a,\,w_1 = b;\,w_n = pw_{n - 1} - qw_{n - 2}\, (n \ge 2)$, where $a$, $b$, $p$ and $q$ are arbitrary…
Recently, the numbers $Y_{n}(\lambda )$ and the polynomials $Y_{n}(x,\lambda)$ have been introduced by the second author [22]. The purpose of this paper is to construct higher-order of these numbers and polynomials with their generating…