  We determine the structure of conformal powers of the Dirac operator on
Einstein {\it 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 Poincar\'e-Einstein metric, and relies on
combinatorial recurrence identities related to the dual Hahn polynomials.
