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

Abstract

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 ∈ Z 2 . From the Lax-pairs we furthermore derive invariants for corresponding reductions to dynamical mappings for some explicit examples.

Multidimensional Toda Lattices: Continuous and Discrete Time

“…which, as was already mentioned, can be considered as the discrete-time Toda equation [46,49]. While on the subject, let us mention that the quotient-difference scheme, along with many other relations between orthogonal polynomials, naturally occur in the context of Padé tables [28].…”

“…The second method is obtained by following the consistency approach from [20] and [49]. In particular, the Lax pair we get for the second method is a certain adaptation of the one from [49] to our setting. The latter approach allows us to present the discrete-time Toda equations (1.13) and the consistency equations (1.7) in a unified fashion in the form of Lax pairs commutation relations, see Subsection 3.4.…”

“…Now we re-derive the dm-Toda equations (2.18) that we already obtained in Subsection 3.4. However, in this case we follow the consistency approach from [20] and [49]. In particular, we get the Lax pair here by using a method that is the adaptation of the one from [49] (see Proposition 3.1).…”

“…However, in this case we follow the consistency approach from [20] and [49]. In particular, we get the Lax pair here by using a method that is the adaptation of the one from [49] (see Proposition 3.1). Since we only consider the case of two measures, we need to consider the family of two measures x t dµ 1 (x) and x t dµ 2 (x), where t ∈ Z + is the discrete time.…”

About several classes of bi-orthogonal polynomials and discrete integrable systems

“…Additionally, in a recent paper [20], Spicer et al have proposed a higher analogue of the discrete-time Toda (HADT) equation…”