The author describes Macdonald’s orthogonal polynomials associated with root systems, observes that for the root system \(BC_ 1\) these are a special case of the Askey-Wilson polynomials, and then finds a generalization of the Macdonald polynomials for \(BC_ n\) that introduces two additional parameters so that when \(n=3D1\) these become the Askey- Wilson polynomials. These generalized Askey-Wilson polynomials are orthogonal with respect to the weight \[ \prod_{\alpha \in R_ 1} {(e^ \alpha;q)_ \infty \over (ae^{\alpha/2}, be^{\alpha/2}, ce^{\alpha/2}, de^{\alpha/2};q)_ \infty} = 20 \prod {(e^ \alpha; q)_ \infty \over (te^ \alpha;q)_ \infty}, \] where \(R_ 1=3D \{\pm 2 \varepsilon_ j \}_{j=3D1,\dots,n}\), \(R_ 2=3D= \{\pm \varepsilon_ i \pm \varepsilon_ j\}_{1 \leq i<j \leq n}\).
For the entire collection see [Zbl 0771.00045].