A semiclassical perspective on multivariate orthogonal polynomials

Abstract

Differential properties for orthogonal polynomials in several variables are studied. We consider multivariate orthogonal polynomials whose gradients satisfy some quasi-orthogonality conditions. We obtain several characterizations for these polynomials including the analogues of the semiclassical Pearson differential equation, the structure relation and a differential-difference equation.

Three term recurrence for the evaluation of multivariate orthogonal polynomials

“…An immediate method to obtain such an explicit formula as described in [8, Theorem 3.2.5] is a left multiplication of (3) by a pseudo-inverse of A n, j and then a rearrangement of terms, cf. [1], but here we aim for more a more explicit form. To that end, we restrict ourselves to the case of the monic orthogonal polynomial vectors m n ∈ Π d n n , formed by m α (x) = x α + q α (x), q α ∈ Π |α|−1 , |α| = n, and note that…”

Two variable Freud orthogonal polynomials and matrix Painlevé-type difference equations

“…Following [2], since the Freud weight function (3.1) is semiclassical, there exists a second order partial differential functional…”

Matrix Pearson Equations Satisfied by Koornwinder Weights in Two Variables

“…The contents of this section are dedicated to recall the definition of the classical and semiclassical character for weight functions in two variables ( [1,2]).…”

“…Observe that, in this case, φ 1 (x) = φ 2 (x) = ρ(x) 2 , the weight function w(x, y) = w 1 (x)w 2 y ρ(x) = (1 − x 2 − y 2 ) α , α > −1, satisfies (15) where…”