On a Symmetric Generalization of Bivariate Sturm–Liouville Problems

Abstract

A new class of partial differential equations having symmetric orthogonal solutions is presented. The general equation is presented and orthogonality is obtained using the Sturm–Liouville approach. Conditions on the polynomial coefficients to have admissible partial differential equations are given. The general case is analyzed in detail, providing orthogonality weight function, three-term recurrence relations for the monic orthogonal polynomial solutions, as well as explicit form of these monic orthogonal polynomial solutions, which are solutions of an admissible and potentially self-adjoint linear second-order partial differential equation of hypergeometric type.

Quadratic Decomposition of Bivariate Orthogonal Polynomials

“…In [15,16] the quadratic decomposition (1.1) has been generalized for polynomial sequences non necessarily symmetric, [13] by using arbitrary polynomials of degree 2 and 1 replacing x 2 and x, respectively, in (1.1), with special attention to the quadratic transformation x 2 − 1 relating Gegenbauer and Jacobi polynomials ( [10]), or even by means of a simple cubic decomposition, as we can read for instance in [5]. Bivariate symmetric orthogonal polynomials as solutions of second-order linear partial difference equations are analysed in [8,20]. In that papers, they also study symmetric generalizations obtaining a new class of partial differential equations having symmetric orthogonal solutions in the bivariate case.…”