Continued Fractions and Hankel Determinants from Hyperelliptic Curves

Abstract

Following van der Poorten, we consider a family of nonlinear maps that are generated from the continued fraction expansion of a function on a hyperelliptic curve of genus g. Using the connection with the classical theory of J -fractions and orthogonal polynomials, we show that in the simplest case g D 1 this provides a straightforward derivation of Hankel determinant formulae for the terms of a general Somos-4 sequence, which were found in a particular form by Chang, Hu, and Xin. We extend these formulae to the higher genus case, and prove that generic Hankel determinants in genus 2 satisfy a Somos-8 relation. Moreover, for all g we show that the iteration for the continued fraction expansion is equivalent to a discrete Lax pair with a natural Poisson structure, and the associated nonlinear map is a discrete integrable system.

Casting light on shadow Somos sequences

“…for all n ∈ Z, which (up to squaring one of the variables) is just the infinite Toda lattice written in Flaschka coordinates. Thus the simultaneous solutions of the map and the Hamiltonian flow provide genus 1 solutions of the Toda lattice, and for genus g > 1 an analogous statement holds for the other maps constructed in [12]; further details will be presented elsewhere. = nx n and overall scale, the third independent shadow solution of Somos-4 can be obtained from the associated solution of the map (3.4), in the form…”

“…It was conjectured by Barry and proved by various authors that certain Somos-4 sequences could be expressed as Hankel determinants [1,2,29]. In [12] we showed that these results can be unified and further generalized by applying van der Poorten's work on Jacobi continued fractions (J-fractions) in hyperelliptic function fields [24]. In the genus 1 case, one expands a certain function G on a quartic curve as a J-fraction, that is…”

“…In order to derive the third shadow solution in a more algebraic way, we consider the following coupled pair of recurrence relations with parameters u, f , which was considered in [12] as the simplest example of a map arising from Jacobi continued fractions in hyperelliptic function fields, based on a construction introduced by van der Poorten [24]:…”

“…of [12], on each level set H = const, the sequence of values of d n coincides with an orbit of (2.4), if the coefficients and the value of the first integral (2.5) are identified as…”

“…3), and obtain four independent shadow sequences. For ease of comparison with Example 4.2 in [12], it is convenient to index the original sequence so that x −1 = x 0 = x 1 = x 2 = 1, and then the first shadow sequence y…”