Orthogonal polynomials in two variables and second-order partial differential equations

We study the second-order partial differential equations L[u] = Auxx +22Buxy + Cuyy + Dux + Euy = λnu, which have orthogonal polynomials in two variables as solutions. By using formal functional calculus on moment functionals, we first give new simpler proofs and improvements of the results by Krall and Sheffer and Littlejohn. We then give a two-variable version of Al-Salam and Chiharaʹs characterization of classical orthogonal polynomials in one variable. We also study in detail the case when L[·] belongs to the basic class, that is, Ay = Cx = 0. In particular, we characterize all such differential equations which have a product of two classical orthogonal polynomials in one variable as solutions.