Recurrence relations of the multi-indexed orthogonal polynomials. IV. Closure relations and creation/annihilation operators

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...

Motzkin Paths, Motzkin Polynomials and Recurrence Relations

We consider the Motzkin paths which are simple combinatorial objects appearing in many contexts. They are counted by the Motzkin numbers, related to the well known Catalan numbers. Associated with the Motzkin paths, we introduce the Motzkin polynomial, which is a multi-variable polynomial “counting” all Motzkin paths of a certain type. Motzkin polynomials (also called Jacobi-Rogers polynomials)...

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 rational functions

It is well known that members of families of polynomials, that are orthogonal with respect to an inner product determined by a nonnegative measure on the real axis, satisfy a three-term recursion relation. Analogous recursion formulas are available for orthogonal Laurent polynomials with a pole at the origin. This paper investigates recursion relations for orthogonal rational functions with arb...