Hankel determinants from quadratic orthogonal pairs for hyperelliptic functions and their applications

As argued by Hone in the paper ( Commun. Pure Appl. Math . 74 (2021), 2310–2347), 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, that is, the bilateral Somos‐4 and Somos‐5 recurrences.