Askey-Wilson polynomials by means of a \(q\)-Selberg type integral
In the literature, several integral representations for orthogonal polynomials are known. As the so-called Askey-Wilson polynomials stand high in the hierarchy of orthogonal polynomials (remember the Askey-tableau and its many limiting procedures to go from one entry to the other), a representation for these polynomials is of independent interest. The main result gives an expression for the \(q\)-Askey-Wilson polynomial \[ p_n(z;a,b,c,d\mid q)=a^{-n}(ab,ac,ad;q)_n {}_4\varphi_3\left( \begin{matrix} q^{-n},q^{-n}abcd,az,az^{-1}\\ ab,ac,ad ;q,q\end{matrix}\right), \] where \((a_1,\ldots,a_m;q)_n=(a_1;q)_n\ldots (a_m;q)_n,\;(a;q)_n=\prod\limits_{0\leq j\leq n-1} (1-aq^j)\) and \({}_4\varphi_3\) is the basic-hypergeometric series \[ {}_4\varphi_3\left(\begin{matrix} a_1,a_2,a_3,a_4 \\ b_1,b_2,b_3; q;z\end{matrix}\right)= \sum_{n=0}^{\infty} {(a_1,a_2,a_3,a_4;q)_n\over (b_1,b_2,b_3,q;q)_n}z^n. \] The author proves \[ {p_n(z;a,b,c,d\mid t)\over (t^{n-1}abcd;t)_n} = \int_{T_n} \prod_{1\leq k\leq n} (z+z^{-1}-x_k-x_k^{-1})\Phi (x) {dx_1\ldots dx_n\over x_1\ldots x_n} \biggl/\int_{T_n} \Phi (x) {dx_1\ldots dx_n\over x_1\ldots x_n}. \] In this \(q\)-Selberg type integral the function \(\Phi(x)=\Phi^{(+)} (x)\Phi^{(+)}(x^{-1})\) is built from \[ \Phi^{(+)}(x)=\prod_{1\leq k\leq n} {(x_k^2;q)_{\infty}\over (ax_k,bx_k,cx_k,dx_k;q)_{\infty}} \prod_{1\leq k<\ell\leq n} {(x_k/x_{\ell},x_kx_{\ell};q)_{\infty}\over (x_k/x_{\ell},tx_kx_{\ell})_{\infty}}, \] where \(\max(|a|,|b|,|c|,|d|)<1\) and \(t>0\). \(\Phi\) is invariant under permutations \(x_j\mapsto x_k\) and transpositions \(x_j\mapsto x_j^{-1}\); this corresponds to the Weyl-group of type \(B_n\). Furthermore \(T^n=\{(x_1,\ldots,x_n)\in{\mathcal C}^n;|x_j|=1\;(1\leq j\leq n)\}\) with counter-clockwise orientation. A highly technical paper that gives in the final section a conjecture on the integral representation for the multivariable MacDonald-Koornwinder polynomials.
scientific article; zbMATH DE number 1417611
scientific article