Related papers: The elliptic function ${\rm dn}_2$ of Shen
Recently, Hong, Mertens, Ono and Zhang proved a conjecture of C\u{a}ld\u{a}raru, He, and Huang that expresses the Taylor series of the modular $j$-function around the elliptic points $i$ and $\rho=e^{\pi i/3}$ as rational functions arising…
In this article we present evaluations of continued fractions studied by Ramanujan. More precisely we give the complete polynomial equations of Rogers-Ramanujan and other continued fractions, using tools from the elementary theory of the…
We define an extension of the Ramanujan trigonometric function to arbitrary dimensions, and give the Dirichlet series generating function. The extension was first given by Eckford Cohen long ago. This links directly to visible point vector…
By using representation theory of the elliptic quantum group U_{q,p}(sl_N^), we present a systematic method of deriving the weight functions. The resultant sl_N type elliptic weight functions are new and give elliptic and dynamical…
As a development of arXiv:1912.12897, we note that the ordinary Shiraishi functions have an insufficient number of parameters to describe generic eigenfunctions of double elliptic system (Dell). The lacking parameter can be provided by…
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…
We introduce and prove evaluations for families of multiple elliptic integrals by solving special types of ordinary and partial differential equations. As an application, we obtain new expressions of Ramanujan-type series of level 4 and…
We give a new derivation and characterisation of the generalised elliptic genus of Krichever-H\"ohn by means of a functional equation.
The purpose of the work is to furnish a complete study of a discrete and special function, discovered by the author and named with the Arabian letter "SHIN" {The letter SHIN is the thirteenth letter of the Arabian alphabet}. It includes…
The classical concept of $Q$-functions associated to symmetric and selfadjoint operators due to M.G. Krein and H. Langer is extended in such a way that the Dirichlet-to-Neumann map in the theory of elliptic differential equations can be…
In 1989 H.Karcher rewrote the theory of elliptic functions through an approach that is much more geometrical than analytical. Therewith he obtained an optimal control over the behaviour and image values of these functions, which allowed for…
We study Neumann functions for divergence form, second order elliptic systems with bounded measurable coefficients in a bounded Lipschitz domain or a Lipschitz graph domain. We establish existence, uniqueness, and various estimates for the…
In this short note we review some facts about elliptic differential operators on Riemannian manifolds.
We suggest a continued fraction origin to Ramanujan's approximation to {(a-b)/(a+b)}^2 in terms of the arc length of an ellipse with semiaxes a and b. Moreover, we discuss the asymptotic accuracy of the approximation.
This is author's Habilitation Thesis (Dr. Sci. dissertation) submitted at the beginning of September 2004. It is written in Russian and is posted due to the continuing requests for the manuscript. The content: 1. Introduction, 2. Nonlinear…
On bounded domains $\Omega \subset \mathbb{R}^d , d \geq 2$, reaching far beyond the scope of Lipschitz domains, we consider an elliptic system of order $2 m$ in divergence form with complex $\mathrm{L}^{\infty}$-coefficients complemented…
Quite recently, McDuff showed that the existence of a symplectic embedding of one four-dimensional ellipsoid into another can be established by comparing their corresponding sequences of ECH capacities. In this note we show that these…
We study estimates of the Green's function in $\mathbb{R}^d$ with $d \ge 2$, for the linear second order elliptic equation in divergence form with variable uniformly elliptic coefficients. In the case $d \ge 3$, we obtain estimates on the…
As a necessary step in constructing elliptic matrix models, which preserve the superintegrability property $<char>\sim {\rm char}$, we suggest an elliptic deformation of the peculiar loci $p_k^{\Delta_n}$, which play an important role in…
We offer a careful development of the Dixonian elliptic functions with parameter $\alpha = 0$ from the initial value problem of which they are solutions.