Remarkable identities related to the (quantum) elliptic Calogero-Sutherland model

We present further remarkable functional identities related to the elliptic Calogero-Sutherland (eCS) system. We derive them from a second quantization of the eCS model within a quantum field theory model of anyons on a circle and at finite temperature. The identities involve two eCS Hamiltonians with arbitrary and, in general, different particle numbers $N$ and $M$, and a particular function of $N+M$ variables arising as anyon correlation function of $N$ particles and $M$ anti-particles. In addition to identities obtained from anyons with the same statistics parameter $\lambda$, we also obtain ``dual'' relations involving ``mixed'' correlation functions of anyons with two different statistics parameters $\lambda$ and $1/\lambda$. We also give alternative, elementary proofs of these identities by direct computations.