Geodesic
“Hidden symmetry” of Askey–Wilson polynomials
Teoretičeskaâ i matematičeskaâ fizika, Tome 89 (1991) no. 2, pp. 190-204 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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 = {{\textquotedblleft}Hidden symmetry{\textquotedblright} of {Askey{\textendash}Wilson} polynomials},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {190--204},
     year = {1991},
     volume = {89},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1991_89_2_a2/}
}
TY  - JOUR
AU  - A. S. Zhedanov
TI  - “Hidden symmetry” of Askey–Wilson polynomials
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 1991
SP  - 190
EP  - 204
VL  - 89
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_1991_89_2_a2/
LA  - ru
ID  - TMF_1991_89_2_a2
ER  - 
%0 Journal Article
%A A. S. Zhedanov
%T “Hidden symmetry” of Askey–Wilson polynomials
%J Teoretičeskaâ i matematičeskaâ fizika
%D 1991
%P 190-204
%V 89
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_1991_89_2_a2/
%G ru
%F 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/

[1] Askey R., Wilson J., SIAM J. Math. Anal., 10:5 (1979), 1008–1016 | DOI | MR | Zbl

[2] Askey R., Wilson J., Mem. Amer. Math. Soc., 54, no. 319, 1985, 1–55 | MR

[3] Vilenkin N. Ya., Spetsialnye funktsii i teoriya predstavlenii grupp, Nauka, M., 1965 | MR

[4] Nikiforov A. F., Suslov S. K., Uvarov V. B., Klassicheskie ortogonalnye polinomy diskretnoi peremennoi, Nauka, M., 1985 | MR

[5] Vaksman L. L., Soibelman Ya. S., Funktsion. analiz i ego pril., 22:3 (1988), 1–14 | MR | Zbl

[6] Koornwinder T. H., Preprint AM-R9013, Amsterdam, 1990 | MR

[7] Atakishiev N. M., Suslov S. K., TMF, 85:1 (1990), 64–73 | MR | Zbl

[8] Granovskii Ya. I., Zhedanov A. S., Izv. vuzov. Fizika, 1986, no. 5, 60–66 | MR

[9] Zhedanov A. S., TMF, 82:1 (1990), 11–17 | MR

[10] Feinsilver P., Acta Appl. Math., 13 (1988), 291–333 | MR | Zbl

[11] Granovskii Ya. I., Zhedanov A. S., Preprint DonFTI-89-7, Donetsk, 1989

[12] Kagramanov E. D. et al., Preprint IC/89/42, Trieste, 1989

[13] Fairlie D. B., J. Phys. A, 23:5 (1990), L183–L187 | DOI | MR | Zbl

[14] Odesskii A. V., Funktsion. analiz i ego pril., 20:2 (1986), 78–79 | MR | Zbl

[15] Sklyanin E. K., Funktsion. analiz i ego pril., 16:4 (1982), 27–34 ; 17:4 (1983), 34–48 | MR | Zbl | MR | Zbl

[16] Granovskii Ya. I., Zhedanov A. S., ZhETF, 94:10 (1988), 49–54 | MR

[17] Granovskii Ya. I., Zhedanov A. S., Lutsenko I. M., ZhETF, 99:2 (1991), 369–377 | MR

[18] Granovskii Ya. I., Zhedanov A. S., Lutzenko I. M., J. Phys. A, 1991 (to appear) | MR

[19] Biedenharn L. C., J. Phys. A, 22:18 (1989), L873–L878 | DOI | MR | Zbl