Koornwinder polynomials

introduction

• The Koornwinder polynomials [43] are a generalisation of the Macdonald polynomials to the root system BCn.
• They depend on six parameters, except for n = 1 when they correspond to the 5-parameter Askey–Wilson polynomials [3].

definition

Throughout this section $x=(x_1,\dots,x_n)$. Then the Koornwinder density is given by \begin{equation}\label{Eq_Kdensity} \Delta(x;q,t;t_0,t_1,t_2,t_3):= \prod_{i=1}^n \frac{(x_i^{\pm 2};q)_{\infty}} {\prod_{r=0}^3 (t_r x_i^{\pm};q)_{\infty}} \prod_{1\leq i<j\leq n} \frac{(x_i^{\pm}x_j^{\pm};q)_{\infty}} {(tx_i^{\pm}x_j^{\pm};q)_{\infty}}, \end{equation} where \begin{align*} (x_i^{\pm};q)_{\infty}&:=(x_i,x_i^{-1};q)_{\infty} \\ (x_i^{\pm}x_j^{\pm};q)_{\infty}&:= (x_ix_j,x_ix_j^{-1},x_i^{-1}x_j,x_i^{-1}x_j^{-1};q)_{\infty}. \end{align*} For complex $q,t,t_0,\dots,t_3$ such that $\lvert{q}\rvert,\lvert{t}\rvert,\lvert{t_0}\rvert,\dots,\lvert{t_3}\rvert<1$ this defines a scalar product on $\mathbb{C}[x^{\pm 1}]$ via \[ \langle{f}{g}\rangle_{q,t;t_0,t_1,t_2,t_3}^{(n)}:= \int_{\mathbb{T}^n} f(x) g(x^{-1})\Delta(x;q,t;t_0,t_1,t_2,t_3) \,d T(x), \] where \[ d T(x):=\frac{1}{2^n n! (2\pi i)^n}\, \frac{d x_1}{x_1}\cdots \frac{d x_n}{x_n}. \] Let $W=\mathfrak{S}_n\ltimes (\Z/2\Z)^n$ be the hyperoctahedral group with natural action on $\mathbb{C}[x^{\pm}]$. For $\lambda$ a partition of length at most $n$, let $m_{\lambda}^W$ be the $W$-invariant monomial symmetric function \[ m_{\lambda}^W(x):=\sum_{\alpha} x^{\alpha} \] summed over all $\alpha$ in the $W$-orbit of $\lambda$. In analogy with the Macdonald polynomials, the Koornwinder polynomials $K_{\lambda}=K_{\lambda}(x;q,t;t_0,t_1,t_2,t_3)$ are defined as the unique family of polynomials in $\Lambda^{\mathrm{BC}_n}:=\mathbb{C}[x^{\pm}]^W$ such that [43] \[ K_{\lambda}=m^W_{\lambda}+\sum_{\mu<\lambda} c_{\lambda\mu} m^W_{\mu} \] and \begin{equation}\label{Eq_KKnul} \langle{K_{\lambda}}{K_{\mu}}\rangle_{q,t;t_0,t_1,t_2,t_3}^{(n)} =0 \qquad\text{if }\lambda\neq\mu. \end{equation} From the definition it follows that the $K_{\lambda}$ are symmetric under permutation of the $t_r$. The quadratic norm was first evaluated in \cite{vDiejen96} (selfdual case) and \cite{Sahi99} (general case). For our purposes we only need \begin{equation}\label{Eq_Gus} \langle{1}{1}\rangle_{q,t;t_0,t_1,t_2,t_3}^{(n)} =\prod_{i=1}^n \frac{(t,t_0t_1t_2t_3t^{n+i-2};q)_{\infty}} {(q,t^i;q)_{\infty}\prod_{0\leq r<s\leq 3}(t_rt_st^{i-1};q)_{\infty}}, \end{equation} known as Gustafson's integral \cite{Gustafson90}.

The $\mathrm{BC}_n$ analogue of the Cauchy identity is given by \cite[Theorem 2.1]{Mimachi01} \begin{align}\label{Eq_Mim} \sum_{\lambda\subseteq m^n} (-1)^{\lvert{\lambda}\rvert} K_{m^n-\lambda}(x;q,t;t_0,t_1,t_2,t_3) K_{\lambda'}(y;t,q;t_0,t_1,t_2,t_3) \\ &=\prod_{i=1}^n\prod_{j=1}^m \big(x_i+x_i^{-1}-y_j-y_j^{-1}\big)\\ &=\prod_{i=1}^n\prod_{j=1}^m x_i^{-1} \big(1-x_iy_j^{\pm}\big), \end{align} where $y=(y_1,\dots,y_m)$ and $(a-b^{\pm}):=(a-b)(a-b^{-1})$.

history

• Several years after the work of Askey and Wilson, Koornwinder extended the Askey–Wilson polynomials to a family of multivariable Laurent polynomials labelled by the non-reduced root system $BC_n$
• The various families of Macdonald (orthogonal) polynomials for classical root systems are all contained in the Koornwinder polynomials, and for a long time it was assumed they represented the highest possible level of generalisation.

related items

• Askey-Wilson polynomials

encyclopedia

• https://en.wikipedia.org/wiki/Koornwinder_polynomials

expositions

• Stokman, Jasper V. “Lecture Notes on Koornwinder Polynomials.” In Laredo Lectures on Orthogonal Polynomials and Special Functions, 145–207. Adv. Theory Spec. Funct. Orthogonal Polynomials. Nova Sci. Publ., Hauppauge, NY, 2004. http://www.ams.org/mathscinet-getitem?mr=2085855.
• Stokman, Jasper V. “Macdonald-Koornwinder Polynomials.” arXiv:1111.6112 [math], November 25, 2011. http://arxiv.org/abs/1111.6112.

articles

• Corteel, Sylvie, and Lauren Williams. ‘Macdonald-Koornwinder Moments and the Two-Species Exclusion Process’. arXiv:1505.00843 [cond-Mat, Physics:nlin], 4 May 2015. http://arxiv.org/abs/1505.00843.
• Stokman, Jasper, and Bart Vlaar. “Koornwinder Polynomials and the XXZ Spin Chain.” Journal of Approximation Theory 197 (September 2015): 69–100. doi:10.1016/j.jat.2014.03.003.
• K. Mimachi, A duality of Macdonald--Koornwinder polynomials and its application to integral representations, Duke Math. J.107 (2001), 265--281.
• Stokman, J. V. “Koornwinder Polynomials and Affine Hecke Algebras.” arXiv:math/0002090, February 11, 2000. http://arxiv.org/abs/math/0002090.
• [43] T. H. Koornwinder, Askey–Wilson polynomials for root systems of type BC in Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications, pp. 189–204, Contemp. Math. 138, Amer. Math. Soc., Providence, 1992. http://oai.cwi.nl/oai/asset/2292/2292A.pdf
• [3] R. Askey and J. A. Wilson, Some basic hypergeometric polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. Vol. 319, AMS, Providence, RI, 1985.