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

Joint eigenfunctions for the relativistic Calogero-Moser Hamiltonians of hyperbolic type. I. First steps