A natural analog would be a representation of the Jack polynomial as an alternating sum of eigenfunctions of the Calogero-Sutherland operator over an orbit of the symmetric group. This can be done even in the q-deformed case (i.e. for Macdonald polynomials and Macdonald operators). Such formula was conjectured by Felder and Varchenko and proved in the paper arXiv:q-alg/9603022, see formula (5-5).
Pavel Etingof
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