  The conical function and its relativistic generalization can be viewed as
eigenfunctions of the reduced 2-particle Hamiltonians of the hyperbolic
Calogero-Moser system and its relativistic generalization. We prove new product
formulas for these functions. As a consequence, we arrive at explicit
diagonalizations of integral operators that commute with the 2-particle
Hamiltonians and reduced versions thereof. The kernels of the integral
operators are expressed as integrals over products of the eigenfunctions and
explicit weight functions. The nonrelativistic limits are controlled by
invoking novel uniform limit estimates for the hyperbolic gamma function.
