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SUMMARY:Continued fractions\, orthogonal polynomials and hyperelliptic cur
 ves - Andrew Hone (University of Kent)
DTSTART:20220923T130000Z
DTEND:20220923T143000Z
UID:TALK180296@talks.cam.ac.uk
DESCRIPTION:After some general remarks on orthogonal polynomials and their
  connection with (discrete) Painleve equations and matrix models\, we cons
 ider a family of nonlinear maps that are generated from the continued frac
 tion expansion of a function on a&nbsp\;hyperelliptic&nbsp\;curve of genus
  g\, as originally described by&nbsp\;van der Poorten. Using the connectio
 n with the classical theory of J-fractions and orthogonal polynomials\, we
  show that in the simplest case g=1 this provides a straightforward deriva
 tion of Hankel determinant formulae for the terms of a general Somos-4 seq
 uence\, which were found in particular cases by Chang\, Hu\, and Xin. We e
 xtend these formulae to the higher genus case\, and prove that generic Han
 kel determinants in genus 2 satisfy a Somos-8 relation. Moreover\, for all
  g we show that the iteration for the continued fraction expansion is equi
 valent to a discrete Lax pair with a natural Poisson structure\, and the a
 ssociated nonlinear map is a discrete integrable system\, connected with s
 olutions of the infinite Toda lattice. If time permits\, we will also ment
 ion the link to S-fractions via contraction\, and a family of maps associa
 ted with the Volterra lattice\, described in current joint work with John 
 Roberts and Pol Vanhaecke.
LOCATION:Seminar Room 2\, Newton Institute
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