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Séminaire Lotharingien de Combinatoire, B72a (2015), 27 pp.

# Tom H. Koornwinder

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

**Abstract.**
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.

Received: August 27, 2014.
Accepted: June 22, 2015.
Final Version: July 17, 2015.

The following versions are available:

**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