Orthogonal polynomials in two variables as solutions of higher order partial differential equations
Let \(L_N[p]\) be a linear partial differential equation of spectral type \[ L_N[p]\equiv \sum_{h=1}^N\sum_{i=0}^ha_{h-i,i}(x,y)\frac{\partial^h}{\partial x^{h-i}\partial y^i}p(x,y)=\lambda_np(x,y) \] with polynomials \(a_{h-i,i}.\) A moment functional \(u\) is a linear functional which is defined on monomials. It is called quasi-definite if there exists a polynomial system \(\{\mathbb P_n\}_{n\geq 0},\) satisfying \[ \langle u,\mathbb P_n\mathbb P_m^t\rangle=0,\quad n\neq m; \qquad\langle u,\mathbb P_n\mathbb P_n^+\rangle=H_n,\quad n=0,1,\dots, \] where \(H_n\) is a symmetric and non-singular matrix, \(\mathbb P_n\) is a \((n+1)-\)vector of polynomials of total degree \(n\) (i.e., \(\mathbb P_n\in\mathcal M_{(n+1)\times 1}(\mathcal P_n))\) independent modulus \(\mathcal P_{n-1}\). A quasi-definite moment functional \(u\) is called a Bochner-Krall moment functional if there exists an operator \(L_N[p]\) and a sequence of matrices \(\Lambda_n\in \mathcal M_{n+1}{\mathbb R}\) such that \(H_n\Lambda_n^t=\Lambda_nH_n\), \(L_N[\mathbb P_n]=\Lambda_n\mathbb P_n\). The corresponding sequence \(\{\mathbb P_n\}\) is called a Bochner-Krall weak orthogonal polynomial system. The authors study Bochner-Krall orthogonal polynomials and their properties. In particular, fourth order partial differential equations as well as some examples are investigated.
scientific article; zbMATH DE number 5840455
scientific article