  The Inozemtsev Hamiltonian is an elliptic generalization of the differential
operator defining the BC_N trigonometric quantum Calogero-Sutherland model, and
its eigenvalue equation is a natural many-variable generalization of the Heun
differential equation. We present kernel functions for Inozemtsev Hamiltonians
and Chalykh-Feigin-Veselov-Sergeev-type deformations thereof. Our main result
is a solution of a heat-type equation for a generalized Inozemtsev Hamiltonian
which is the source for all these kernel functions. Applications are given,
including a derivation of simple exact eigenfunctions and eigenvalues for the
Inozemtsev Hamiltonian.
