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

Orthogonal polynomial eigenfunctions of second-order partial differerential equations

“…We will always assume that |A| + |B| + |C| ≡ 0 since otherwise equation (2.1) cannot have any OPS as solutions (see [3]). Following Krall and Sheffer [7], we also assume that equation (2.1) is admissible, that is, λ m = λ n for m = n (or equivalently an + g = 0 for each n ≥ 0) so that equation (2.1) has a unique monic PS of solutions.…”

“…We are concerned with the problem raised by Krall and Sheffer [7] (see also [3], [4], [11]): classify all second-order partial differential equations of the type (1.3)…”

Two variable orthogonal polynomials and structured matrices

“…Special examples of these types of polynomials have arisen in studies related to symmetry groups (Dunkl [3], Koornwinder [14], MacDonald [20]), as extensions of one variable polynomials (Fernández-Pérez-Piñar [5], Koornwinder [13]) and as eigenfunctions of partial differential equations (Koornwinder [12], Krall-Sheffer [17], Kim-Kwon-Lee [11], Kwon-Lee-Littlejohn [19] (see also the references in [4])). The general theory of these polynomials can trace its origins back to Jackson [10] and an excellent review of the theory can be found in the book of Dunkl and Xu [4] (see also the book of Suetin [21]).…”

Classical orthogonal polynomials in two variables: a matrix approach

“…In fact, [8] is the starting point to prove results for orthogonal polynomials in several variables similar to the standard properties for one-variable orthogonal polynomials (see [3,5,6,15]). …”