Characterization of classical orthogonal polynomials in two variables
Maurice Kenfack Nangho, Kerstin Jordaan, Bleriod Jiejip Nkwamouo
TL;DR
Problem: characterize two-variable orthogonal polynomials in the sense of a weight ρ by a unified set of equivalent properties. Approach: prove the equivalence among the matrix Pearson equation div($\rho$ $\Phi$) = $\rho(\psi_1,\psi_2)$, a second-order PDE, gradient orthogonality, a Rodrigues-type formula $\mathbb{P}_n^t = (-1)^n/\rho \; div^{(n)}(\rho \Phi^{\otimes n}) \nabla^{(n)} X_n^t \prod_{j=0}^{n-1} \Lambda_{n,j}^{-1}$, and a structure relation with matrices $\Lambda_{n,j}$ and $A^{n,m}_p$, establishing their mutual equivalence. Key contributions include a precise COP definition in two variables, a characterization theorem with explicit Rodrigues and structure-relations, and a triangle-domain example with computable $\Lambda_{n,j}$ and PDE correspondence. Significance: provides a robust analytical-algebraic framework for two-variable COP, enabling applications in probability, PDEs, and mathematical physics, and clarifying connections with moment-functionals.
Abstract
For a family of polynomials in two variables, orthogonal with respect to a weight function, we prove under some conditions, equivalence between: the matrix Pearson equation of the weight, the second order linear partial differential equation, the orthogonality of the gradients, the matrix Rodrigues formula involving tensor product of matrices, and the so-called first structure relation. We then introduce a notion of classical orthogonal polynomials in two variables and relate the corresponding theory for weight functions and moment functionals. Finally, we present a nontrivial example that illustrates and delineates our contribution to the field.