Polynomials of the Askey Scheme as Clebsch-Gordan Coefficients

Abstract

Given a semi-simple algebra equipped with a coproduct, the Clebsch-Gordan coefficients are the elements of the transition matrices between direct product representation and its irreducible decomposition. It is well known that the Clebsch-Gordan coefficients of the Lie algebra $\mathfrak{sl}_2$ are given in terms of the dual Hahn polynomials. Taking the reversed point of view, we show that any finite dimensional family of polynomials belonging to the Askey scheme can be interpreted as Clebsch-Gordan coefficients of an algebra. The Hahn polynomials are thus associated to the oscillator algebra with the Krawtchouk polynomials treated through a limit. The dual Hahn polynomials and Racah polynomials are seen to be associated to $\mathfrak{sl}_2$ with a more general coproduct than the standard one. The $q$-Hahn polynomials are interpreted as Clebsch-Gordan coefficients of a $q$-deformation of the oscillator algebra and the $q$-Racah polynomials are seen to be connected in this way to ${\rm U}_q(\mathfrak{sl}_2)$ with a generalized coproduct.