Bernstein-type bases on $q$-quadratic lattices and Askey--Wilson\slash $q$-Racah connection coefficients

Abstract

On the $q$-quadratic lattices that carry the Askey--Wilson polynomials, the two roles played by a single Bernstein basis on the linear lattice, namely being a nonnegative partition of unity and carrying an orthogonal connection with the classical families, split between two distinct bases. The affine (spectral) basis retains the Bézier properties. A first-order lattice ladder exists in degrees at most two, but fails in degree three for every $q$, as shown by an explicitly factorised determinant. The affine factors do not satisfy the weight-shift mechanism that produces the $q$-Racah connection for the generalized-power basis. The generalized-power basis, built from products of two Askey--Wilson monomials, does not furnish a nonnegative partition of unity in a natural positive parameter region, specified in the text; there the unique normalisation summing to unity has sign-changing coefficients. Nevertheless, its connection coefficients with the Askey--Wilson polynomials are identified completely: they are $q$-Racah polynomials $R_m(μ(k);\,ad/q,\,bc/q,\,q^{-n-1},\,a/b\,|\,q)$ multiplied by fully explicit prefactors. The proof is bispectral, valid for every degree, and rests on the tridiagonal action of the Askey--Wilson operator on the generalized-power basis, with explicit band coefficients. A scaled limit recovers the Wilson/Racah connection, while a second limit gives the $q$-Hahn polynomial part of the normalised $q$-linear connection coefficients. A numerical case study on the NACA 2412 airfoil illustrates the orthogonal machinery.