Identities for Rankin–Cohen brackets, Racah coefficients and associativity

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 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.