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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.

Joint Eigenfunctions for the Relativistic Calogero–Moser Hamiltonians of Hyperbolic Type: I. First Steps