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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Continued fractions\, orthogonal polynomials and h
yperelliptic curves - Andrew Hone (University of K
ent)
DTSTART;TZID=Europe/London:20220923T140000
DTEND;TZID=Europe/London:20220923T153000
UID:TALK180296AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/180296
DESCRIPTION:After some general remarks on orthogonal polynomia
ls and their connection with (discrete) Painleve e
quations and matrix models\, we consider a family
of nonlinear maps that are generated from the cont
inued fraction expansion of a function on a \;
hyperelliptic \;curve of genus g\, as original
ly described by \;van der Poorten. Using the c
onnection 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 f
ound in particular cases by Chang\, Hu\, and Xin.
We extend these formulae to the higher genus case\
, and prove that generic Hankel determinants in ge
nus 2 satisfy a Somos-8 relation. Moreover\, for a
ll 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 a
ssociated nonlinear map is a discrete integrable s
ystem\, connected with solutions of the infinite T
oda lattice. If time permits\, we will also mentio
n the link to S-fractions via contraction\, and a
family of maps associated with the Volterra lattic
e\, described in current joint work with John Robe
rts and Pol Vanhaecke.
LOCATION:Seminar Room 2\, Newton Institute
CONTACT:
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