Related papers: Bilinear recurrences and addition formulae for hyp…

Let $V$ be a hyperelliptic curve of genus 2 defined by $Y^2=f(X)$, where $f(X)$ is a polynomial of degree 5. The sigma function associated with $V$ is a holomorphic function on $\mathbb{C}^2$. For a point $P$ on $V$, we consider the problem…

We define and study a recurrence relation in ${\mathbb Z}^3$, called the hexahedron recurrence, which is similar to the octahedron recurrence (Hirota bilinear difference equation) and cube recurrence (Miwa equation). Like these examples,…

Hinted by the elliptic parameterization of the Ising model, the addition formula of the elliptic function forms to give the integrable SU(2) group relation in the previous paper. We then expect that the addition formula of the Abelian…

We exhibit a three parameter infinite family of quadratic recurrence relations inspired by the well known Somos sequences. For one infinite subfamily we prove that the recurrence generates an infinite sequence of integers by showing that…

Motivated by the search for an appropriate notion of a cluster superalgebra, incorporating Grassmann variables, Ovsienko and Tabachnikov considered the extension of various recurrence relations with the Laurent phenomenon to the ring of…

We compute the divisor of the modular equation on the modular curve $\Gamma_0(N) \backslash \mathbb H^*$ and then find recurrence relations satisfied by the modular traces of the Hauptmodul for any congruence subgroup $\Gamma_0(N)$ of genus…

We develop the theory of Abelian functions defined using a tetragonal curve of genus six, discussing in detail the cyclic curve $y^4 = x^5 + \lambda_4x^4 + \lambda_3x^3 + \lambda_2x^2 + \lambda_1x + \lambda_0$. We construct Abelian…

We give conjectures on the form of families of integer sequences whose Hankel transforms are, respectively, $(\alpha, \beta)$ Somos $4$ sequences, $(\alpha, 0, \gamma)$ Somos $6$ sequences, and $(\alpha, \beta, \gamma, \delta)$ Somos $8$…

A Somos sequence of order $n$ is defined by a quadratic recurrence of width $n + 1$. Some of the remarkable properties of these sequences for small $n$ are tied to certain matrices built out of them being of finite rank. We give an…

In 1997 the present authors published a review (Ref. BEL97 in the present manuscript) that recapitulated and developed classical theory of Abelian functions realized in terms of multi-dimensional sigma-functions. This approach originated by…

In this article we prove an analogue of a theorem of Lachaud, Ritzenthaler, and Zykin, which allows us to connect invariants of binary oc-tics to Siegel modular forms of genus 3. We use this connection to show that certain modular…

We consider the generalized dual transformation for elliptic/hyperelliptic $\wp$ functions up to genus three. For the genus one case, from the algebraic addition formula, we deduce that the Weierstrass $\wp$ function has the SO(2,1) $\cong$…

We show that any graph polynomial from a wide class of graph polynomials yields a recurrence relation on an infinite class of families of graphs. The recurrence relations we obtain have coefficients which themselves satisfy linear…

We have given full algebraic addition formulae of genus two hyperelliptic functions by the duplication method. This full addition formulae according to the duplication method give some hints of Lie group structure in addition formulae of…

Effective reconstruction formulas of a curve from its theta hyperplanes are known classically in genus 2 (where the theta hyperplanes are Weierstrass points), and 3 (where, for a generic curve, the theta hyperplanes are bitangents to a…

In this paper we provide an explicit construction of a $distinctive$ multiple Dirichlet series associated to products of quadratic Dirichlet L-series, which we believe should be tightly connected to a generalized metaplectic Whittaker…

We develop our method to prove quantum superintegrability of an integrable 2D system, based on recurrence relations obeyed by the eigenfunctions of the system with respect to separable coordinates. We show that the method provides rigorous…

We present new addition formulae for the Weierstrass functions associated with a general elliptic curve. We prove the structure of the formulae in n-variables and give the explicit addition formulae for the 2- and 3-variable cases. These…

It is known that the elliptic function solutions of the nonlinear Schr\"odinger equation are reduced to the algebraic differential relation in terms of the Weierstrass sigma function, $\displaystyle{…

We construct families of hyperelliptic curves over Q of arbitrary genus g with (at least) g integral elements in K_2. We also verify the Beilinson conjectures about K_2 numerically for several curves with g=2, 3, 4 and 5. The paper is…