A direct approach to the bispectral problem for the Ruijsenaars-Macdonald q-difference operators

Abstract

We present a direct approach to the bispectral problem associated with the Ruijsenaars-Macdonald q-difference operators of GL type. We give an explicit construction of the meromorphic function psi_n(x;s|q,t) on T^n_x T^n_s, which is the solution of the bispectral problem up to a certain gauge transformation. Some basic properties of psi_n(x;s|q,t) are studied, including the structure of the divisor of poles, the symmetries psi_n(x;s|q,t)=psi_n(s;x|q,t)=psi_n(x;s|q,q/t), and the recursive structure described in terms of Jackson integrals or q-difference operators.