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arXiv · 1005.0482

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

Paul E. Spicer·Frank W. Nijhoff·Peter H. van der Kamp
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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.

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Paul E. Spicer, Frank W. Nijhoff, Peter H. van der Kamp. 2011-02-21. Higher analogues of the discrete-time Toda equation and the quotient-difference algorithm. https://doi.org/10.1088/0951-7715%2F24%2F8%2F006

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