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

Xiang-Ke Chang, Jiyuan Liu

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.

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

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.
Paper Structure (18 sections, 18 theorems, 189 equations)

This paper contains 18 sections, 18 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. On J-fractions and Hankel determinants
  3. Continued fraction expansions in
  4. Hankel determinant representations for J-fractions
  5. Continued fractions and quadratic orthogonal pairs for hyperelliptic functions
  6. Proper quadratic extensions over
  7. Hyperelliptic functions of genus 1
  8. Laurent series expansions
  9. Continued fraction expansions
  10. Quadratic orthogonal pairs in
  11. Hankel determinant expressions
  12. Applications to bilateral Somos-4 and Somos-5
  13. Connection formulae
  14. Initial value problem for Somos-4
  15. Initial value problem for Somos-5
  16. ...and 3 more sections

Key Result

Theorem 2.1

Suppose $\langle 0,\alpha_1X+\beta_1,\alpha_2X+\beta_2,\ldots \rangle = s_0 X^{-1} +s_1 X^{-2}+\cdot\cdot\cdot$, then, for any $n \in \mathbb{N}^*$, there hold where $\alpha_0$, $\Delta_{n}$, $\Delta_{n}^*$ are given by

Theorems & Definitions (43)

  • Theorem 2.1
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Remark 3.4
  • Theorem 3.5
  • proof
  • Remark 3.6
  • ...and 33 more