# Higher analogues of the discrete-time Toda equation and the quotient-difference algorithm - Nonlinear Sciences > Exactly Solvable and Integrable Systems

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Abstract: The discrete-time Toda equation arises as a universal equation for therelevant Hankel determinants associated with one-variable orthogonalpolynomials through the mechanism of adjacency, which amounts to the inclusionof shifted weight functions in the orthogonality condition. In this paper weextend this mechanism to a new class of two-variable orthogonal polynomialswhere the variables are related via an elliptic curve. This leads to a `Higherorder Analogue of the Discrete-time Toda- HADT equation for the associatedHankel determinants, together with its Lax pair, which is derived from therelevant recurrence relations for the orthogonal polynomials. In a similar wayas the quotient-difference QD algorithm is related to the discrete-time Todaequation, a novel quotient-quotient-difference QQD scheme is presented forthe 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 forcorresponding reductions to dynamical mappings for some explicit examples.

Autor: Paul E. Spicer, Frank W. Nijhoff, Peter H. van der Kamp

Fuente: https://arxiv.org/