Related papers: Jacobi's epsilon and zeta function for moduli outs…
In this paper we obtain a set of five new transmutations of the mother formula. Further, we obtain the second set of ten exact metafunctional equations by crossbreeding on every two elements of the previous set. Elements of the last set…
We extend some results recently obtained by Dan Romik about the Taylor coefficients of the theta function $\theta_{3}\left(1\right)$ to the case $\theta_{3}\left(q\right)$ of an arbitrary value of the elliptic modulus $k.$ These results are…
This paper discuss a new class of functional equations by using both Poisson summation formula and Jacobi type theta a function. The class of Riemann type functional equations are derived from self-reciprocal probability density functions.…
In this article we present ways to evaluate certain sums, products and continued fractions using tools from the theory of elliptic functions. The specific results appear to be new, although similar ones can be found in the leterature; in…
The Jacobian elliptic functions are generalized to functions including the generalized trigonometric functions. The paper deals with the basis property of the sequence of generalized Jacobian elliptic functions in any Lebesgue space. In…
We present some new results in theory of classical theta-functions of Jacobi and sigma-functions of Weierstrass: ordinary differential equations (dynamical systems) and series expansions. The paper is basically organized as a stream of new…
In this short paper, we find the transformation formula for the theta series under the action of the Jacobi modular group on the Siegel-Jacobi space. This formula generalizes the formula (5.1) obtained by Mumford in his book[p.189, Tata…
We look for differential equations satisfied by the generalized Jacobi polynomials which are orthogonal on the interval [-1,1] with respect to a weight function consisting of the classical Jacobi weight function together with point masses…
In this paper two identities involving a function defined by the complete elliptic integrals of the first and second kinds are proved. Some functional inequalities and elementary estimates for this function are also derived from the…
Classical Jacobi polynomials $P_{n}^{(\alpha,\beta)}$, with $\alpha, \beta>-1$, have a number of well-known properties, in particular the location of their zeros in the open interval $(-1,1)$. This property is no longer valid for other…
We show how Jacobian elliptic functions (JEF) can be used to solve ordinary differential equations (ODE) describing nonlinear dynamics of microtubules (MT). We demonstrate that only one of JEFs can be used while the remaining two do not…
We define "values" of the elliptic modular j-function at real quadratic irrationalities by using Hecke's hyperbolic Fourier expansions, and present some observations based on numerical experiments.
Properties of the Jacobi Theta3-function and its derivatives under discrete Fourier transforms are investigated, and several interesting results are obtained. The role of modulo N equivalence classes in the theory of Theta-functions is…
The well-known Jacobi elliptic functions sn(z)$, $cn(z), dn(z) are defined in higher dimensional spaces by the following method. Consider the Clifford algebra of the antieuclidean vector space of dimension 2m+1. Let x be the identity…
In this paper we establish a close connection between three notions at- tached to a modular subgroup. Namely the set of weight two meromorphic modular forms, the set of equivariant functions on the upper half-plane commuting with the action…
An error estimate for the Gauss-Lobatto quadrature formula for integration over the interval $[-1, 1]$, relative to the Jacobi weight function $w^{\alpha,\beta}(t)=(1-t)^\alpha(1+t)^\beta$, $\alpha,\beta>-1$, is obtained. This estimate…
We introduce two explicit examples of polynomials orthogonal on the unit circle. Moments and the reflection coefficients are expressed in terms of Jacobi elliptic functions. We find explicit expression for these polynomials in terms of a…
We look for spectral type differential equations satisfied by the generalized Jacobi polynomials, which are orthogonal on the interval [-1,1] with respect to a weight function consisting of the classical Jacobi weight function together with…
In this paper, we study modularity of several functions which naturally arose in a recent paper of Lau and Zhou on open Gromov-Witten potentials of elliptic orbifolds. They derived a number of examples of indefinite theta functions, and we…
The aim of this paper is to show how zeta functions and excision in cyclic cohomology may be combined to obtain index theorems. In the first part, we obtain a local index formula for "abstract elliptic pseudodifferential operators"…