“Hidden symmetry” of Askey–Wilson polynomials

A new $q$-commutator Lie algebra with three generators, $AW(3)$, is considered, and its finite-dimensional representations are investigated. The overlap functions between the two dual bases in this algebra are expressed in terms of Askey–Wilson polynomials of general form of a discrete argument: to the four parameters of the polynomials there correspond four independent structure parameters of the algebra. Special and degenerate cases of the algebra $AW(3)$ that generate all the classical polynomials of discrete arguments – Racah, Hahn, etc., – are considered. Examples of realization of the algebra $AW(3)$ in terms of the generators of the quantum algebras of $SU(2)$ and the $q$-oscillator are given. It is conjectured that the algebra $AW(3)$ is a dynamical symmetry algebra in all problems in which $q$-polynomials arise as eigenfunctions.