  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.
