Minimal Solutions of Three-Term Recurrence Relations and Orthogonal Polynomials

Asymptotics of Orthogonal Polynomials via Recurrence Relations

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...

Asymptotic behavior of solutions of general three term recurrence relations

We consider the solutions of general three term recurrence relations whose coefficients are analytic functions in a prescribed region. We study the ratio asymptotic of such solutions under the assumption that the coefficients are asymptotically periodic and their strong asymptotic under more restrictive conditions.

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.

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...