On sieved orthogonal polynomials. X. General blocks of recurrence relations

Asymptotics of Orthogonal Polynomials via Recurrence Relations

Recurrence Relations for Orthogonal Polynomials on Triangular Domains

Abstract: In Farouki et al, 2003, Legendre-weighted orthogonal polynomials Pn,r(u, v, w), r = 0, 1, . . . , n, n ≥ 0 on the triangular domain T = {(u, v, w) : u, v, w ≥ 0, u+ v+w = 1} are constructed, where u, v, w are the barycentric coordinates. Unfortunately, evaluating the explicit formulas requires many operations and is not very practical from an algorithmic point of view. Hence, there is...

Orthogonal Polynomials Defined by a Recurrence Relation

R. Askey has conjectured that if a system of orthogonal polynomials is defined by the three term recurrence relation xp„-,(x) = -^ p„(x) + an_xPn_x(x) + -^pn-2(x) In tn-\ and (-0 then the logarithm of the absolutely continuous portion of the corresponding weight function is integrable. The purpose of this paper is to prove R. Askey's conjecture...

General block bi - orthogonal polynomials

We discuss formal orthogonal polynomials with respect to a moment matrix that has no structure whatsoever. In the classical case the moment matrix is often a Hankel or a Toeplitz matrix. We link this to block factorization of the moment matrix and its inverse, the block Hessenberg matrix of the recurrence relation, the computation of successive Schur complements and general subspace iterative m...

Recurrence Formulas for Multivariate Orthogonal Polynomials

In this paper, necessary and sufficient conditions are given so that multivariate orthogonal polynomials can be generated by a recurrence formula. As a consequence, orthogonal polynomials of total degree n in d variables that have dim n¡( common zeros can now be constructed recursively. The result is important to the construction of Gaussian cubature formulas.