Degenerate Sklyanin algebras, Askey–Wilson polynomials and Heun operators

Abstract

Abstract The q-difference equation, the shift and the contiguity relations of the Askey–Wilson polynomials are cast in the framework of the three and four-dimensional degenerate Sklyanin algebras s k a 3 and s k a 4 . It is shown that the q-para Racah polynomials corresponding to a non-conventional truncation of the Askey–Wilson polynomials form a basis for a finite-dimensional representation of s k a 4 . The first order Heun operators defined by a degree raising condition on polynomials are shown to form a five-dimensional vector space that encompasses s k a 4 . The most general quadratic expression in the five basis operators and such that it raises degrees by no more than one is identified with the Heun–Askey–Wilson operator.