Rational Ruijsenaars-Schneider hierarchy and bispectral difference operators

Abstract

We show that a monic polynomial in a discrete variable $n$, with coefficients depending on time variables $t_1,t_2,...$ is a $\tau$-function for the discrete Kadomtsev-Petviashvili hierarchy if and only if the motion of its zeros is governed by a hierarchy of Ruijsenaars-Schneider systems. These $\tau$-functions were considered in [12], where it was proved that they parametrize rank one solutions to a difference-differential version of the bispectral problem.

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