二元二阶线性偏微分方程与正交多项式解
Bivariate second--order linear partial differential equations and orthogonal polynomial solutions
摘要：
本文构造了二元二阶超几何型线性偏微分方程的容许潜自伴正交多项式解的主要代数和微分性质及权函数。所有这些性质的一般公式得到明确的偏微分方程的多项式系数，使用向量矩阵符号。此外，还给出了多项式本征解及其任意阶偏导数的Rodrigues表示.最后，作为例子，这些结果被应用到特定的Appell和Koornwinder正交多项式，同一偏微分方程的解决方案。
英文摘要：
In this paper we construct the main algebraic and differential properties and the weight functions of orthogonal polynomial solutions of bivariate second--order linear partial differential equations, which are admissible potentially self--adjoint and of hypergeometric type. General formulae for all these properties are obtained explicitly in terms of the polynomial coefficients of the partial differential equation, using vector matrix notation. Moreover, Rodrigues representations for the polynomial eigensolutions and for their partial derivatives of any order are given. Finally, as illustration, these results are applied to specific Appell and Koornwinder orthogonal polynomials, solutions of the same partial differential equation.
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