Following van der Poorten, 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. 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 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. © 2020 the Authors. Communications on Pure and Applied Mathematics is published by Wiley Periodicals LLC.

中文翻译：

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超椭圆曲线的连分数和汉克尔行列式

继 van der Poorten 之后，我们考虑一系列非线性映射，这些映射是由函数在g属的超椭圆曲线上的连分数展开生成的。使用与J分数和正交多项式的经典理论的联系，我们表明在最简单的情况下g = 1这提供了一般 Somos-4 序列项的 Hankel 行列式公式的直接推导，这些公式在Chang、Hu 和 Xin 的特殊形式。我们将这些公式扩展到更高的属情况，并证明属 2 中的通用 Hankel 行列式满足 Somos-8 关系。此外，对于所有g我们表明连分数展开的迭代等效于具有自然泊松结构的离散 Lax 对，并且相关的非线性映射是离散可积系统。© 2020 作者。《纯数学与应用数学通讯》由 Wiley Periodicals LLC 出版。