q二次格上的伯恩斯坦型基与Askey–Wilson/q-Racah连接系数
Bernstein-type bases on $q$-quadratic lattices and Askey--Wilson\slash $q$-Racah connection coefficients
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中文总结 AI 辅助
该研究在承载Askey–Wilson多项式的q二次格上拆分了线性格伯恩斯坦基的双重角色，确定了广义幂基与Askey–Wilson多项式的q-Racah型连接系数，并通过翼型案例阐释了正交机制。
中文摘要 AI 辅助
在承载Askey–Wilson多项式的q二次格上，线性格上单个伯恩斯坦基所扮演的两个角色——即作为非负单位分拆以及承载与经典族的正交连接——被拆分为两个不同的基。仿射（谱）基保留了贝塞尔（Bézier）性质；一阶格梯算子存在于次数不超过2的情况，但对于任意q，在次数为3时失效，这一点可通过显式因式分解的行列式证明。仿射因子不满足能为广义幂基生成q-Racah连接的权重移位机制。由两个Askey–Wilson单项式的乘积构建的广义幂基，在文本指定的自然正参数区域内无法提供非负单位分拆，该区域内求和为1的唯一归一化系数存在符号变化；不过，其与Askey–Wilson多项式的连接系数已被完全确定：它们是q-Racah多项式$R_m(\boldsymbol{\mu}(k);\\,ad/q,\\,bc/q,\\,q^{-n-1},\\,a/b\\,|\\,q)$乘以完全显式的 prefactors（前置因子）。该证明具有双谱性，对任意次数均成立，且依赖于Askey–Wilson算子在广义幂基上的三对角作用，带有显式带系数；缩放极限可恢复Wilson/Racah连接，而第二个极限给出归一化q线性连接系数的q-Hahn多项式部分。针对NACA 2412翼型的数值案例研究阐释了该正交机制。
英文摘要
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 $q\in(0,1)$. 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.
发表机构
- IFCAE, Departamento de Matemática Aplicada II, E.E. Aeronáutica e do Espazo, Universidade de Vigo（西班牙维戈大学航空航天工程学院应用数学二系）
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