Non-autonomous multidimensional Toda system and multiple interpolation problem

TL;DR

This paper explores a non-autonomous multidimensional Toda system related to Hermite-Padé approximation, providing determinant solutions and linking it to integrable discrete systems like Hirota's discrete KP equations.

Contribution

It introduces a novel non-autonomous multidimensional Toda system as a reduction of Hirota's discrete KP equations, with explicit determinant solutions and connections to Hermite-Padé approximation.

Findings

Determinant solution for the interpolation problem

Reduction to non-autonomous discrete-time Toda equations in dimension two

Connection established between approximation problems and integrable systems

Abstract

We study the interpolation analogue of the Hermite-Pad\'e type I approximation problem. We provide its determinant solution and we write down the corresponding integrable discrete system as an admissible reduction of Hirota's discrete Kadomtsev-Petviashvili equations. Apart from the $\tau$-function form of the system we provide its variant, which in the simplest case of dimension two reduces to the non-autonomous discrete-time Toda equations.