Recursion and explicit formulas for particular N-variable Knop-Sahi and Macdonald polynomials

Knop and Sahi simultaneously introduced a Family of non-homogeneous, non-symmetric polynomials, G(alpha)(x; q, t). The top homogeneous components of these polynomials are the non-symmetric Macdonald polynomials, E-alpha(x; q, t). Pin appropriate Hecke algebra symmetrization of E-alpha yields the Macdonald polynomials, P-lambda(x; q, t). A search for explicit formulas for the polynomials G(alpha)(x; q, t) led to thr main results of this paper. In particular, we give a complete solution for the case G((k, a, ...,))(x; q, t). A remarkable by-product of our proofs is the discovery that these polynomials satisfy a recursion on the number of variables. (C) 1999 Academic Press.