Hankel Determinants from Quadratic Orthogonal Pairs for Hyperelliptic Functions and Their Applications

TL;DR

This paper introduces quadratic orthogonal pairs to resolve a longstanding mismatch problem in Hankel determinants from hyperelliptic functions, and applies these results to initial value problems of discrete integrable systems.

Contribution

It presents a new concept called quadratic orthogonal pairs for hyperelliptic functions, solving an open problem and enabling applications to discrete integrable systems.

Findings

Resolved the mismatch problem in Hankel determinants from hyperelliptic curves.

Provided a thorough treatment of initial value problems for bilateral Somos-4 and Somos-5 recurrences.

Abstract

As argued by Hone in the paper [Commun. Pure Appl. Math., 74(11):2310--2347, 2021], a ``mismatch" problem remained unsolved while he was investigating continued fraction expansions and Hankel determinants from hyperelliptic curves. In this paper, by introducing a new notion called quadratic orthogonal pairs for hyperelliptic functions, we resolve the corresponding problem. As further applications, we give a thorough treatment of the initial value problems for two discrete integrable systems, i.e. the bilateral Somos-4 and Somos-5 recurrences.