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 coprimality argument completely classifies the first-order lattice ladder: its solution space has dimensions two and one in degrees one and two, respectively, and is trivial in every degree at least three for every q∈(0,1). 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.
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BibTeX
@misc{area2026bernsteintype,
title = {{Bernstein-type bases on $q$-quadratic lattices and Askey--Wilson\slash $q$-Racah connection coefficients}},
author = {Iván Area},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.04802},
eprint = {2608.04802},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.04802},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bernstein-type bases on q-quadratic lattices and Askey--Wilson\slash q-Racah connection coefficients
AU - Iván Area
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2608.04802
UR - https://arxiv.org/abs/2608.04802
AB - 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 coprimality argument completely classifies the first-order lattice ladder: its solution space has dimensions two and one in degrees one and two, respectively, and is trivial in every degree at least three for every q∈(0,1). 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.
ER -