Bivariate version of the Hahn-Sonine theorem

Abstract

Abstract. We consider orthogonal polynomials in two variables whose derivatives with respect to x are orthogonal. We show that they satisfy a system of partial differential equations of the form) is a vector of polynomials in x and y for n ≥ 0, and Λn is an eigenvalue matrix of order (n + 1) × (n + 1) for n ≥ 0. Also we obtain several characterizations for these polynomials. Finally, we point out that our results are able to cover more examples than Bertran's.

Sobolev orthogonality of polynomial solutions of second-order partial differential equations

“…Moreover, note that u (s) 0 and v (t) satisfy the sufficient conditions of Theorem 4.7 with α = μ − 1/2 and b(t) = 1 − t 2 . Indeed, it is well known that w μ also satisfies the diagonal Pearson equations (Álvarez de Morales et al 2009b;Lee 2000;Marcellán et al 2018a, b)…”