Rational Ruijsenaars–Schneider hierarchy and bispectral difference operators

We show that a monic polynomial in a discrete variable n , with coefficients depending on time variables t 1 , t 2 , … , is a τ -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 τ -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.