Polynomials of the Askey scheme as Clebsch-Gordan coefficients

TL;DR

This work establishes a reverse correspondence between Clebsch–Gordan coefficients and the ($q$-)Askey scheme of polynomials, showing that any finite polynomial family can serve as CG coefficients for an appropriately defined algebra with a coproduct. By leveraging contiguity relations, the authors construct algebras (and their coproducts) so that the CG decomposition of tensor products reproduces chosen polynomials, yielding realizations for the oscillator algebra with Hahn/Krawtchouk limits and for $sl_2$ with dual Hahn and Racah extensions, as well as their $q$-analogs in $osc_q$ and $U_q(sl_2)$. The approach unifies disparate objects—orthogonal polynomials, representation theory, and quantum groups—providing explicit coproducts and highlighting non-coassociativity as an intrinsic feature, potentially resolvable via Drinfeld twists. These results enrich the interplay between special functions and algebraic structures, offering a framework to explore higher-order CG symbols and their functional counterparts in future work.

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 $sl_2$ 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 $sl_2$ 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 $U_q(sl_2)$ with a generalized coproduct.