Orthogonal polynomials in two variables and second-order partial differential equations
The authors consider the second order partial differential equations of spectral type of the form \[ L[u]:= Au_{xx}+ 2Bu_{xy}+ Cu_{yy}+ Du_x+ Eu_y= \lambda_nu, \qquad n=0,1,2,\dots\;, \tag{\(*\)} \] where \(A(x,y),\dots, E(x,y)\) are polynomials, independent of \(n\), while \(\lambda_n\) is the eigenvalue parameter, and find a characterization of orthogonal polynomials satisfying it. In the particular case when \(A_y= C_x=0\), they characterize \((*)\) which have a product of two classical orthogonal polynomials in one variable as solutions. Furthermore, they give conditions under which derivatives of any orthogonal polynomial solution to \((*)\) are also orthogonal polynomials satisfying the same type of equations as \((*)\). Notice that some of the results are already known but the proofs given here by the authors, which are based on the use of formal functional calculus on movement functionals, are simpler than those given elsewhere [see for example: \textit{H. L. Krall} and \textit{I. M. Sheffer}, Ann. Math. Pura Appl., IV. Ser. 325-376 (1967; Zbl 0186.38602)].
scientific article; zbMATH DE number 1084081