SOME CLASSICAL EXPANSIONS FOR KNOP-SAHI AND MACDONALD POLYNOMIALS

In recent simultaneous work, Knop and Sahi introduced a non-homogeneous non-symmetric polynomial G (x;q;t) whose highest homogeneous component gives the non-symmetric Macdonald polynomial E (x;q;t). Macdonald shows that for any compo- sition that rearranges to a partition , an appropriate Hecke algebra symmetrization of E yields the Macdonald polynomial P (x;q;t). In the original papers all these polyno- mials are only shown to exist. No explicit expressions are given relating them to the more classical bases. Our basic discovery here is that G (x;q;t) appears to have surprisingly elegant expansions in terms of the polynomials Z (x1;:::;xn;q) = Qn i=1(xi;q) i. In this paper we present the rst results obtained in the problem of determining the connec- tion coecients relating these bases. In particular we give a solution to the problem of two variables. Our proofs rely on the theory of basic hypergeometric series and reveal a deep connection between this classical subject and the theory of Macdonald polynomials. n n , they form themselves a basis for the polynomials in x1;x2;:::;xn. In view of the fundamental nature of the polynomials P (x;q;t) and their central place in symmetric function theory, it is reasonable to assume that the E should also play a central role in the study of polynomials in several variables. It is shown in (12) that in fact theE themselves are but a special case of families of orthogonal polynomials associated to root systems; the E being associated the root system An. As for the P , the E have only been shown to exist. They have 1