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Continued fractions, orthogonal polynomials and hyperelliptic curves

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ARA2 - Applicable resurgent asymptotics: towards a universal theory

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.

This talk is part of the Isaac Newton Institute Seminar Series series.

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