Related papers: Transformation Formula of the "2nd" Order Mock The…

We prove an $\text{SL}_2 (\mathbb{Z})$-invariance property of multivariable trace functions on modules for a regular VOA. Applying this result, we provide a proof of the inversion transformation formula for Siegel theta series. As another…

We define generalised zeta functions associated to indefinite quadratic forms of signature (g-1,1) -- and more generally, to complex symmetric matrices whose imaginary part has signature (g-1,1) -- and we investigate their properties. These…

In this work at first the relation the Mittag-Lefler function to the exponential is given. The results are applied to the construction of the solution of Cauchy problem for ordinary linear operator differential equations with constant…

We present a method for reducing the order of ordinary differential equations satisfying a given scaling relation (Majorana scale-invariant equations). We also develop a variant of this method, aimed to reduce the degree of non-linearity of…

We derive formulae which lend themselves to TQFT interpretations of the Milnor torsion, the Lescop invariant, the Casson invariant, and the Casson-Morita cocyle of a 3-manifold, and, furthermore, relate them to the Reshetikhin-Turaev…

Let $\Delta= \sum_{m=0}^\infty q^{(2m+1)^2} \in \mathbb{F}_2[[q]]$ be the reduction mod 2 of the $\Delta$ series. A modular form $f$ modulo $2$ of level 1 is a polynomial in $\Delta$. If $p$ is an odd prime, then the Hecke operator $T_p$…

In this paper we utilize the covariance of Ricatti equation with respect to linear fractional transformations to define classes of conformally equivalent second order differential equations. This motivates then the introduction of…

We study the SU(2) Witten--Reshetikhin--Turaev invariant for the Seifert fibered homology spheres with M-exceptional fibers. We show that the WRT invariant can be written in terms of (differential of) the Eichler integrals of modular forms…

Recently, an interesting dilogarithmic integral arising in quantum field theory has been closed-form evaluated in terms of the Clausen function $\text{Cl}_2(\theta)$ by Coffey [J. Math. Phys.} 49 (2008), 093508]. It represents the volume of…

In this paper, we study the symmetry of quantum torus with the concept of crossed product algebra. As a classical counterpart, we consider the orbifold of classical torus with complex structure and investigate the transformation property of…

We compute the second moment of the Dedekind zeta function of a quadratic field times an arbitrary Dirichlet polynomial of length $T^{1/11-\epsilon}$.

To any algebraic variety X and and closed 2-form \omega on X, we associate the "symplectic action functional" T(\omega) which is a function on the formal loop space LX introduced by the authors in math.AG/0107143. The correspondence \omega…

Addition formulas for theta functions of arbitrary order are shown and applied to the theoretical understanding of the fractional quantum Hall effect in a multi-layer two-dimensional many-electron system under periodic conditions.

The classical transformation of Jacobi's theta function admits a simple proof by producing an integral representation that yields this invariance apparent. This idea seems to have first appeared in the work of S. Ramanujan. Several examples…

The Fourier transform of a bounded measurable function, $f$, on the real line is shown to be the second distributional derivative of a H\"older continuous function. The Fourier transform is written as the difference of $\int_{-1}^1…

We present a theta function representation of the twisted characters for the rational N=2 superconformal field theory, and discuss the Jacobi-form like functional properties of these characters for a fixed central charge under the action of…

We exhibit an infinite family of vector-valued mock theta functions indexed by positive integers coprime to $6$. These are built from specializations of Dyson's rank generating function and related functions studied by Watson, Gordon, and…

Properties of four quintic theta functions are developed in parallel with those of the classical Jacobi null theta functions. The quintic theta functions are shown to satisfy analogues of Jacobi's quartic theta function identity and…

In this paper, we give an extension of the classical story of the elliptic modular function to the Hilbert modular case for $\mathbb{Q}(\sqrt{5})$. We construct the period mapping for a family $\mathcal{F}=\{S(X,Y)\}$ of $K3 $ surfaces with…

We propose an integral transform, called metamorphism, which allow us to reduce the order of a differential equation. For example, the second order Helmholtz equation is transformed into a first order equation, which can be solved by the…