(t,q) Q-systems, DAHA and quantum toroidal algebras via generalized Macdonald operators

We introduce difference operators on the space of symmetric functions which are a natural generalization of the $(q,t)$-Macdonald operators. In the $t\to\infty$ limit, they satisfy the $A_{N-1}$ quantum $Q$-system. We identify the elements in the spherical $A_{N-1}$ DAHA which are represented by these operators, as well as within the quantum toroidal algebra of $gl_1$ and the elliptic Hall algebra. We present a plethystic, or bosonic, formulation of the generating functions for the generalized Macdonald operators, which we relate to recent work of Bergeron et al. Finally we derive constant term identities for the current that allow to interpret them in terms of shuffle products. In particular we obtain in the $t\to\infty$ limit a shuffle presentation of the quantum $Q$-system relations.