Abstract. We determine the structure of conformal powers of the Dirac operator on Einstein Spin-manifolds in terms of the product formula for shifted Dirac operators. The result is based on the techniques of higher variations for the Dirac operator on Einstein manifolds and spectral analysis of the Dirac operator on the associated Poincaré-Einstein metric, and relies on combinatorial recurrence identities related to the dual Hahn polynomials.

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In this paper we introduce a new family of operator-valued distributions on Euclidian space acting by convolution on differential forms. It provides a natural generalization of the important Riesz distributions acting on functions, where the corresponding operators are (−∆) −α/2 , and we develop basic analogous properties with respect to meromorphic continuation, residues, Fourier transforms, and relations to conformal geometry and representations of the conformal group.2010 Mathematics Subject Classification. 46F10; 47B06, 31B10, 53A30.