The rational Sklyanin algebra and the Wilson and para-Racah polynomials

TL;DR

This paper explores the connection between Wilson and para-Racah polynomials and the rational Sklyanin algebra, revealing new algebraic structures and operators related to these polynomials.

Contribution

It establishes a novel link between the rational Sklyanin algebra and specific orthogonal polynomials, introducing new operators and finite-dimensional representations.

Findings

Derived second order Heun operators on quadratic grids.

Linked Wilson and para-Racah polynomials to algebra representations.

Identified rational degenerations of the Sklyanin algebra.

Abstract

The relation between Wilson and para-Racah polynomials and representations of the degenerate rational Sklyanin algebra is established. Second order Heun operators on quadratic grids with no diagonal terms are determined. These special or S-Heun operators lead to the rational degeneration of the Sklyanin algebra; they also entail the contiguity and structure operators of the Wilson polynomials. The finite-dimensional restriction yields a representation that acts on the para-Racah polynomials.