AI 中文总结
研究单位三角形上与特定权重相关的对称正交多项式，构造对称正交基，证明其为微分算子特征函数，得到基的显式表示与平方范数，确定相关线性偏微分算子代数，该代数同构于两变量实多项式环。
AI 中文摘要
我们研究了单位三角形上一族与权重\[ w_{\alpha,\gamma,\kappa}(x,y) = (xy)^\alpha(1 - x - y)^\gamma |x - y|^{2\kappa + 1}, \quad \alpha,\gamma,\kappa > -1,\quad \alpha+\kappa>-\frac32 \]相关的对称正交多项式。我们在单纯形腔上构造了相应的首一对称正交基，并证明其元素是形式自伴二阶微分算子\(\mathcal D_1^{\alpha,\gamma,\kappa}\)的特征函数。一对伴随阶梯算子产生了具有相同特征函数的二阶算子\(\mathcal D_2^{\alpha,\kappa}\)。我们还根据单变量雅可比多项式得到了基的显式表示并计算了其平方范数。转换到初等对称变量后，我们确定了以变换后的多项式为特征函数的实多项式系数线性偏微分算子的完整代数。每个这样的算子都可以唯一地写成\(\mathcal D_1^{\alpha,\gamma,\kappa}\)和\(\mathcal D_2^{\alpha,\kappa}\)的多项式；因此，这个代数同构于两个变量的实多项式环。
英文摘要
We study a family of symmetric orthogonal polynomials on the unit triangle associated with the weight \[ w_{α,γ,κ}(x,y) = (xy)^α(1-x-y)^γ|x-y|^{2κ+1}, \quad α,γ,κ>-1,\quad α+κ>-\frac32. \] We construct the corresponding monic symmetric orthogonal basis on the simplex chamber and prove that its elements are eigenfunctions of a formally self-adjoint second-order differential operator \(\mathcal D_1^{α,γ,κ}\). A pair of adjoint ladder operators yields a second operator of order two \(\mathcal D_2^{α,κ}\) with the same eigenfunctions. We also obtain an explicit representation of the basis in terms of one-variable Jacobi polynomials and compute its squared norms. After passing to the elementary symmetric variables, we determine the full algebra of linear partial differential operators with real polynomial coefficients having the transformed polynomials as eigenfunctions. Every such operator can be written uniquely as a polynomial in \(\mathcal D_1^{α,γ,κ}\) and \(\mathcal D_2^{α,κ}\); consequently, this algebra is isomorphic to the real polynomial ring in two variables.
