``Hidden symmetry'' of Askey--Wilson polynomials
Teoretičeskaâ i matematičeskaâ fizika, Tome 89 (1991) no. 2, pp. 190-204
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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.
@article{TMF_1991_89_2_a2,
author = {A. S. Zhedanov},
title = {``Hidden symmetry'' of {Askey--Wilson} polynomials},
journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
pages = {190--204},
publisher = {mathdoc},
volume = {89},
number = {2},
year = {1991},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TMF_1991_89_2_a2/}
}
A. S. Zhedanov. ``Hidden symmetry'' of Askey--Wilson polynomials. Teoretičeskaâ i matematičeskaâ fizika, Tome 89 (1991) no. 2, pp. 190-204. http://geodesic.mathdoc.fr/item/TMF_1991_89_2_a2/