Abstract
AbstractFollowing 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.
Highlights
1; 1; 1; 1; 2; 3; 7; 23; 59; 314; 1529; 8209; 83313; 620297; : : : : A proof of this fact was eventually published [38], but a better understanding of the mechanism by which such rational recurrences can yield integer sequences came from the observation that (1.1) exhibits the Laurent property [21, 22]: the iterates are Laurent polynomials in the initial values with integer coefficients, that is to say n 2 Z 01 ; 11 ; 21 ; 31 ; ; Communications on Pure and Applied Mathematics, 0001–0038 (PREPRINT)
The latter result does not overlap with that of Xin, since the conditions on the coefficients and initial conditions do not include the original sequence (1.2). It was subsequently shown by Chang, Hu, and Xin that, for any Somos-4 sequence with two adjacent initial values equal to 1, the terms with positive index n are given by a Hankel determinant of the form (1.4), where the entries zsj satisfy a recursion of the form (1.8), for a suitable choice of z ; z; z; zs0 ; zs1 [11]
We show how the nonlinear map coming from the continued fraction arises from a Poisson map on this phase space, which preserves the same Hamiltonians and Casimirs as the continuous system
Summary
D 1 and the four initial values 0 ; 1 ; 2 ; 3 are all 1, the recurrence (1.1) generates a sequence of integers [51], beginning with (1.2). This underlying algebraic structure has many consequences, including the existence of higher-order relations between the terms [26, 36, 48] and more refined versions of the Laurent property that produce large families of integer sequences [30] From this point of view, Somos-4 sequences are natural extensions of Ward’s elliptic divisibility sequences [57], which correspond to the special case P0 D 1 (the identity element in the group law of E ), and generalize the arithmetical properties of Fibonacci or Lucas sequences to a nonlinear setting [16]. D1;:::;n ; where the entries zsj are obtained from the function D .x/ satisfying the alge-
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