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, Gα(x; q, t). The top homogeneous components of these polynomials are the non-symmetric Macdonald polynomials, Eα(x; q, t). An appropriate Hecke algebra symmetrization of Eα yields the Macdonald polynomials, Pλ(x; q, t). A search for explicit formulas for the polynomials Gα(x; q, t) led to the main results of this paper. In particular, we give a complete solution for the case G(k,a,...,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.