Lifted Koornwinder polynomials

introduction

$ \newcommand{\la}{\lambda} \newcommand{\La}{\Lambda} $

• The lifted Koornwinder polynomials $\tilde{K}_{\lambda}$ are a $7$-parameter family of inhomogeneous symmetric functions
• They are invariant under permutations of the $t_r$ and form a $\mathbb{Q}(q,t,T,t_0,t_1,t_2,t_3)$ basis of $\Lambda$.
• As a function of the $t_r$ the lifted Koornwinder polynomial $\tilde{K}_{\la}$ has poles at

\begin{equation}\label{Eq_poles} t_0t_1t_2t_3=q^{2-\la_i-j} t^{i+\la'_j}T^{-2}, \qquad (i,j)\in\la. \end{equation}

• Via the binomial formula, the lifted BCn interpolation polynomials lead immediately to a lifting for Koornwinder polynomials

definition

The lifted Koornwinder polynomials are defined by the expansion \[ \tilde{K}_\lambda(;q,t,T;t_0,t_1,t_2,t_3) = \sum_{\mu\subset\lambda} {\lambda \brack \mu}_{q,t,(T/t)\sqrt{t_0t_1t_2t_3/q}} \frac{k^0_\lambda(q,t,T;t_0{:}t_1,t_2,t_3)} {k^0_\mu(q,t,T;t_0{:}t_1,t_2,t_3)} \tilde{P}^*_\mu(;q,t,T;t_0). \] Here $\tilde{P}^*_\mu(;q,t,T;t_0)$ denotes Lifted BCn interpolation polynomials

• this is analogous to the following formula for Koornwinder polynomials

$$ K_{\lambda}(x;q,t;t_0,t_1,t_2,t_3) =\sum_{\mu\subseteq\lambda} {\lambda \brack \mu}_{q,t,s} \, \frac{K_{\lambda}\big(t_0(1,t,\dots,t^{n-1});q,t;t_0,t_1,t_2,t_3\big)} {K_{\mu}\big(t_0(1,t,\dots,t^{n-1});q,t;t_0,t_1,t_2,t_3\big)}\, \bar{P}_{\mu}^{\ast(n)}(x;q,t,t_0), $$

Koornwinder polynomial

• Let $K^{(n)}_\lambda$ be Koornwinder polynomials

thm

For any integer $n>0$ and partition $\lambda$, and for generic values of the parameters, \[ \tilde{K}_\lambda(x_1^{\pm 1},\dots x_n^{\pm 1};q,t,t^n;t_0,t_1,t_2,t_3) = \begin{cases} K^{(n)}_\lambda(x_1,\dots x_n;q,t;t_0,t_1,t_2,t_3) & \ell(\lambda)\le n\\ 0 & \text{otherwise.} \end{cases} \]

examples

• For example, $\tilde{K}_0=1$ and

\[ \tilde{K}_1(q,t,T;t_0,t_1,t_2,t_3)= m_1+\frac{1-T}{(1-t)(1-t_0t_1t_2t_3T^2/t^2)} \sum_{r=0}^3 \Big(\frac{t_0t_1t_2t_3T}{t_rt}-t_r\Big). \]

related items

• Koornwinder polynomials

articles

• Rains, Eric M. “BCn-Symmetric Polynomials.” Transformation Groups 10, no. 1 (March 2005): 63–132. doi:10.1007/s00031-005-1003-y. http://arxiv.org/abs/math/0112035.