“Hidden symmetry” of Askey-Wilson polynomials

Conclusions

We have shown that the Askey-Wilson polynomials of general form are generated by the algebra AW(3), which has a fairly simple structure and is the q-analog of a Lie algebra with three generators. The main properties of these polynomials (weight function, recursion relation, etc.) can be obtained directly from analysis of the representations of the algebra.

In this paper, we have considered finite-dimensional representations of the algebra AW(3) and the Aksey-Wilson polynomials of discrete argument corresponding to these representations. A separate analysis is required for the infinite-dimensional representations, which generate polynomials of a continuous argument (these polynomials were investigated in detail in the review [2]). Also of interest is investigation of representations of the algebra AW(3) for complex values of the basic parameter ω and of the structure parameters.

In our view, the algebra AW(3) by itself warrants careful study on account of several remarkable properties (in the first place, the duality with respect to the operators K₀, K₁) not present in the currently very popular quantum algebras of the type SU_q(2).

We assume that the algebra AW(3) is an algebra of dynamical or “hidden” symmetry in all problems in which exponential or hyperbolic spectra and the corresponding q-polynomials arise. We hope that in time the algebra AW(3) will come to play the same role in “q-problems” as Lie algebras play in exactly solvable problems of quantum mechanics.