Higher analogues of the discrete-time Toda equation and the quotient-difference algorithm

TL;DR

This paper extends the discrete-time Toda equation to a higher-order analogue involving two-variable orthogonal polynomials on elliptic curves, introducing a new integrable scheme and analyzing its properties.

Contribution

It introduces the Higher order Analogue of the Discrete-time Toda (HADT) equation and a novel quotient-quotient-difference (QQD) scheme related to elliptic curve orthogonal polynomials.

Findings

Derivation of the HADT equation and its Lax pair.

Development of the QQD scheme for the HADT equation.

Existence of well-posed s-periodic initial value problems.

Abstract

The discrete-time Toda equation arises as a universal equation for the relevant Hankel determinants associated with one-variable orthogonal polynomials through the mechanism of adjacency, which amounts to the inclusion of shifted weight functions in the orthogonality condition. In this paper we extend this mechanism to a new class of two-variable orthogonal polynomials where the variables are related via an elliptic curve. This leads to a `Higher order Analogue of the Discrete-time Toda' (HADT) equation for the associated Hankel determinants, together with its Lax pair, which is derived from the relevant recurrence relations for the orthogonal polynomials. In a similar way as the quotient-difference (QD) algorithm is related to the discrete-time Toda equation, a novel quotient-quotient-difference (QQD) scheme is presented for the HADT equation. We show that for both the HADT equation and the QQD scheme, there exists well-posed $s$-periodic initial value problems, for almost all $\s\in\Z^2$. From the Lax-pairs we furthermore derive invariants for corresponding reductions to dynamical mappings for some explicit examples.