Okounkov's BC-Type Interpolation Macdonald Polynomials and Their q=1 Limit

Tom H. Koornwinder

Séminaire lotharingien de combinatoire, Tome 72 (2014-2015)

Résumé

This paper surveys eight classes of polynomials associated with A-type and BC-type root systems: Jack, Jacobi, Macdonald and Koornwinder polynomials and interpolation (or shifted) Jack and Macdonald polynomials and their BC-type extensions. Among these the BC-type interpolation Jack polynomials were probably unobserved until now. Much emphasis is put on combinatorial formulas and binomial formulas for (most of) these polynomials. Possibly new results derived from these formulas are a limit from Koornwinder to Macdonald polynomials, an explicit formula for Koornwinder polynomials in two variables, and a combinatorial expression for the coefficients of the expansion of BC-type Jacobi polynomials in terms of Jack polynomials which is different from Macdonald's combinatorial expression. For these last coefficients in the two-variable case the explicit expression of Koornwinder and Sprinkhuizen [SIAM J. Math. Anal. 9 (1978), 457--483] is now obtained in a quite different way.

Comment by the author. There are a few unfortunate misprints in the article. These are:

formula (10.7):
in second line:
in summation range m_1+m_2 -> m_1-m_2
(q^{-m_1+m_2};q)_j -> (q^{-m_1+m_2};q)_{j+k}
in third line:
(q^{m_2}ax_1,q^{m_2}ax_1^{-1};q)_k -> (q^{m_2}ax_2,q^{m_2}ax_2^{-1};q)_k
The corrected formula can be read in http://arxiv.org/abs/1408.5993.

formula (10.14):
second upper parameter of the 3F2: t -> \tau