2026/08/05 by Iván Area
Mathematics · #math.CA #msc:33D45 #msc:33C45 #msc:33D15 #msc:41A10 #msc:42C05 #msc:65D17
3 figures
arxiv created 2026/08/05 · arxiv updated 2026/08/06
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 Rm(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.