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Continued fractions and Hankel determinants from hyperelliptic curves

Hone, Andrew N.W. (2020) Continued fractions and Hankel determinants from hyperelliptic curves. Communications on Pure and Applied Mathematics, . ISSN 0010-3640. (doi:10.1002/cpa.21923) (KAR id:79346)

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Abstract

Following van der Poorten, we consider a family of nonlinear maps which 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 = 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 two 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.

Item Type: Article
DOI/Identification number: 10.1002/cpa.21923
Uncontrolled keywords: J-fraction, hyperelliptic curve, Hankel determininant, Somos sequence
Subjects: Q Science > QA Mathematics (inc Computing science)
Divisions: Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science
Depositing User: Andrew Hone
Date Deposited: 18 Dec 2019 15:09 UTC
Last Modified: 03 Mar 2021 14:45 UTC
Resource URI: https://kar.kent.ac.uk/id/eprint/79346 (The current URI for this page, for reference purposes)
Hone, Andrew N.W.: https://orcid.org/0000-0001-9780-7369
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