The Askey scheme as a four-manifold with corners

TL;DR

This paper geometrically interprets the Askey scheme of orthogonal polynomials as a four-manifold with corners, providing a continuous parametrization that links different polynomial families through boundary restrictions.

Contribution

It introduces a reparametrization of Racah and Wilson polynomials to form a manifold with corners, unifying various orthogonal polynomial families within the Askey scheme.

Findings

Reparametrization ensures continuity in parameters.

Restriction to zero yields lower families in the scheme.

Geometric interpretation as a manifold with corners.

Abstract

Racah and Wilson polynomials with dilated and translated argument are reparametrized such that the polynomials are continuous in the parameters as long as these are nonnegative, and such that restriction of one or more of the new parameters to zero yields orthogonal polynomials lower in the Askey scheme. Geometrically this will be described as a manifold with corners.