Rational Ruijsenaars–Schneider hierarchy and bispectral difference operators

We show that a monic polynomial in a discrete variable nn, with coefficients depending on time variables t1,t2,…t1,t2,…, is a ττ-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 ττ-functions were considered in [L. Haine, P. Iliev, Commutative rings of difference operators and an adelic flag manifold, Int. Math. Res. Not. 2000 (6) (2000) 281–323], where it was proved that they parametrize rank one solutions to a difference–differential version of the bispectral problem.