Ladder operators for bivariate generalized classical symmetric orthogonal polynomials

Abstract

Classical generalized bivariate polynomials are families of bivariate symmetric polynomials pγ n,k(x, y) orthogonal with respect to the weight function Wγ(x, y) = ω(x)ω(y)|x − y| 2γ+1, x, y ∈ (a, b), where γ > −1, and ω(t) is one of the classical weight functions (Hermite, Laguerre, Jacobi) on the real line. They are eigenfunctions of Dγ 1 , a second order partial differential with rational coefficients. We consider raising or lowering operators for these polynomials, that is, we study differential operators acting on orthogonal polynomials to raise or lower their degree while preserving their orthogonality but shifting the parameters in the weight function. The change of variables u = x+y, v = xy allows us to construct a family of orthogonal polynomials by means of the identity qγ n,k(u, v) = pγ n,k(x, y), those polynomials are eigenfunctions of partial differential operators with polynomial coefficients and order 2 and 4 constructed from Dγ 1 and the ladder operators. Finally, we show that these two operators generate the algebra of differential operators that admit the polynomials qγ n,k(u, v) as eigenfunctions.