  We prove an infinite family of identities satisfied by the Rankin-Cohen
brackets involving the Racah polynomials. A natural interpretation in the
representation theory of sl(2) is provided. From these identities and known
properties of the Racah polynomials follows a short new proof of the
associativity of the Eholzer product. Finally, we discuss, in the context of
Rankin-Cohen algebras introduced by D.Zagier, how any algebraic identity
satisfied by the Rankin-Cohen brackets can be seen as a consequence of the set
of identities presented in this paper.
