  We present and develop a recursion scheme to construct joint eigenfunctions
for the commuting analytic difference operators associated with the integrable
N-particle systems of hyperbolic relativistic Calogero-Moser type. The scheme
is based on kernel identities we obtained in previous work. In this first paper
of a series we present the formal features of the scheme, show explicitly its
arbitrary-N viability for the `free' cases, and supply the analytic tools to
prove the joint eigenfunction properties in suitable holomorphy domains.
