Abstract

After some general remarks on orthogonal polynomials and their connection with (discrete) Painleve equations and matrix models, 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, as originally described by van der Poorten. Using the connection with the classical theory of J-fractions and orthogonal polynomials, we show that in the simplest case g=1 this provides a straightforward derivation of Hankel determinant formulae for the terms of a general Somos-4 sequence, which were found in particular cases 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, connected with solutions of the infinite Toda lattice. If time permits, we will also mention the link to S-fractions via contraction, and a family of maps associated with the Volterra lattice, described in current joint work with John Roberts and Pol Vanhaecke.