Identities for Rankin-Cohen brackets, Racah coefficients and associativity

Abstract

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.