On the evaluation formula for Jack polynomials with prescribed symmetry

The Jack polynomials with prescribed symmetry are obtained from the nonsymm etric polynomials via the operations of symmetrization, anti symmetrization and normalization. After dividing out the corresponding antisymmetric poly nomial of smallest degree, a symmetric polynomial results. Of interest in a pplications is the value of the latter polynomial when all the variables ar e set equal. Dunkl has obtained this evaluation, making use of a certain sk ew-symmetric operator. We introduce a simpler operator for this purpose, th ereby obtaining a new derivation of the evaluation formula. An expansion fo rmula of a certain product in terms of Jack polynomials with prescribed sym metry implied by the evaluation formula is used to derive a generalization of a constant term identity due to Macdonald, Kadell and Kaneko. Although w e do not give the details in this paper, the operator introduced here can b e defined for any reduced crystallographic root system, and used to provide an evaluation formula for the corresponding Heckman-Opdam polynomials with prescribed symmetry.