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A higher rank generalization of the (rank one) Racah algebra is obtained asthe symmetry algebra of the Laplace-Dunkl operator associated to the$\mathbb{Z}_2^n$ root system. This algebra is also the invariance algebra ofthe generic superintegrable model on the $n$-sphere. Bases of Dunkl harmonicsare constructed explicitly using a Cauchy-Kovalevskaia theorem. These basesconsist of joint eigenfunctions of maximal Abelian subalgebras of the higherrank Racah algebra. A method to obtain expressions for both the connectioncoefficients between these bases and the action of the symmetries on thesebases is presented.

A higher rank Racah algebra and the $\mathbb{Z}_2^{n}$ Laplace-Dunkl operator