Lifted Koornwinder polynomials
introduction
Via the binomial formula, the lifted 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, \[ \tilde{K}_\lambda(x_1^{\pm 1},\dots x_n^{\pm 1};q,t,t^n;t_0,t_1,t_2,t_3) = \begin{cases} K^{(n)}_\lambda(x_1,\dots x_n;q,t;t_0,t_1,t_2,t_3) & \ell(\lambda)\le n\\ 0 & \text{otherwise.} \end{cases} \]