Recurrence relations of the multi-indexed orthogonal polynomials. III

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...

A formula for the coefficients of orthogonal polynomials from the three-term recurrence relations

In this work, the coefficients of orthogonal polynomials are obtained in closed form. Our formula works for all classes of orthogonal polynomials whose recurrence relation can be put in the form Rn(x) = x Rn−1(x) − αn−2 Rn−2(x). We show that Chebyshev, Hermite and Laguerre polynomials are all members of the class of orthogonal polynomials with recurrence relations of this form. Our formula unif...

Recurrence Relations for Orthogonal Polynomials and Algebraicity of Solutions of the Dirichlet Problem

We show that any finite-term recurrence relation for planar orthogonal polynomials in a domain imply that the domain must be an ellipse. Our proof relies on Schwarz function techniques and on elementary properties of functions in Sobolev spaces.

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...