On orthogonal polynomials with perturbed 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...

Comparative Asymptotics for Perturbed Orthogonal Polynomials

Let {Φn}n∈N0 and {Φ̃n}n∈N0 be such systems of orthonormal polynomials on the unit circle that the recurrence coefficients of the perturbed polynomials Φ̃n behave asymptotically like those of Φn. We give, under weak assumptions on the system {Φn}n∈N0 and the perturbations, comparative asymptotics as for Φ̃n(z)/Φ ∗ n(z) etc., Φ ∗ n(z) := z Φ̄n( 1 z ), on the open unit disk and on the circumference ma...

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...

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.