Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-28T00:02:16.608Z Has data issue: false hasContentIssue false
  • English
  • Français

Addition Formula For Big q-Legendre Polynomials From The Quantum Su(2) Group

Published online by Cambridge University Press:  20 November 2018

H. T. Koelink
H. T. Koelink*
Affiliation:
Department of Mathematics Katholieke Universiteit Leuven Celestijnenlaan200 B B-3001 Leuven (Heverlee) Belgium e–mail: erik.koelink@wis.KULeuven.ac.be
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

From Koornwinder's interpretation of big q-Legendre polynomials as spherical elements on the quantum SU(2) group an addition formula is derived for the big g-Legendre polynomial. The formula involves Al-Salam-Carlitz polynomials, little q-Jacobi polynomials and dual q-Krawtchouk polynomials. For the little q-ultraspherical polynomials a product formula in terms of a big q-Legendre polynomial follows by q-integration. The addition and product formula for the Legendre polynomials are obtained when q tends to 1.

Keywords

33D8033D4533C4542C05addition formulaproduct formulabig q-Legendre polynomialslittle q-Jacobi polynomialsAl-Salam-Carlitz polynomialsquantum groupsSU(2)g-Charlier polynomialsLegendre polynomials.
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 47 , Issue 2 , 01 April 1995 , pp. 436 - 448
Copyright
Copyright © Canadian Mathematical Society 1995

References

1. Al-Salam, W.A. and Carlitz, L., Some orthogonal q-polynomials, Math. Nachr. 30(1965), 47–61.Google Scholar
2. Andrews, G.E. and Askey, R., Enumeration of partitions: The role of Eulerian series and q-orthogonal polynomials. In: Higher Combinatorics, (ed. Aigner, M.), Reidel, 1977. 3–26.Google Scholar
3. Andrews, G.E., Classical orthogonal polynomials, In: Polynômes Orthogonaux et Applications, Lecture Notes Math. 1171, (eds. Brezinski, C., Draux, A., Magnus, A.R., Maroni, R. and Ronveaux, A.), Springer, 1985. 36–62.Google Scholar
4. Askey, R., Orthogonal Polynomials and Special Functions, CBMS-NSF Regional Conf. Ser. in Appl. Math. 21,SIAM, 1975.Google Scholar
5. Askey, R. and Wilson, J., A set of orthogonal polynomials that generalize the Racah coefficients or 6 — j symbols, SIAM J. Math. Anal. 10(1979), 1008–1016.Google Scholar
6. Chihara, T.S., An Introduction to Orthogonal Polynomials, Math. Appl. 13, Gordon and Breach, 1978.Google Scholar
7. Gasper, G. and Rahman, M., Basic Hypergeometric Series, Encyclopedia Math. Appl. 35, Cambridge University Press, 1990.Google Scholar
8. H, M.E.. Ismail and Wilson, J.A., Asymptotic and generating relations for the q-Jacobi and 4ϕ3 polynomials, J. Approx. Theory 36(1982), 43–54.Google Scholar
9. Koelink, H.T., H ansen-Lommel orthogonality relations for Jackson's q-Bessel functions, J. Math. Anal. Appl. 175(1993), 425–437.Google Scholar
10. Koelink, H.T., The addition formula for continuous q-Legendre polynomials and associated spherical elements on the SU(2) quantum group related to Askey-Wilson polynomials, SIAM J. Math. Anal. 25(1994), 197–217.Google Scholar
11. Koornwinder, T.H., Krawtchouk polynomials, a unification of two different group theoretic interpretations, SIAM J. Math. Anal. 13(1982), 1011–1023.Google Scholar
12. Koornwinder, T.H., Orthogonal polynomials in connection with quantum groups. In: Orthogonal Polynomials: Theory and Practice, NATO Adv. Sci. Inst. Ser. C 294, (ed. Nevai, P.), Kluwer, 1990. 257–292.Google Scholar
13. Koornwinder, T.H., The addition formula for little q-Legendre polynomials and the SU(2) quantum group, SIAM J. Math. Anal. 22(1991), 195–301.Google Scholar
14. Koornwinder, T.H., Askey-Wilson polynomials as zonal spherical functions on the SU(2) quantum group, SIAM J. Math. Anal. 24(1993), 795–813.Google Scholar
15. Nikiforov, A.F. and Uvarov, V.B., Special Functions of Mathematical Physics, Translated from the Russian by R.Boas, P., Birkhàuser, 1988.Google Scholar
16. Noumi, M., Quantum groups and q-orthogonal polynomials. Towards a realization of Askey-Wilson polynomials on SUq(2). In: Special Functions, ICM-90 Satell. Conf. Proc, (eds. Kashiwara, M. and Miwa, T.), Springer, 1991. 260–288.Google Scholar
17. Noumi, M. and Mimachi, K., Askey-Wilson polynomials and the quantum group SUq(2), Proc. Japan Acad. Ser. A 66(1990), 146–149.Google Scholar
18. , Askey-Wilson polynomials as spherical functions on SUq(2). In: Quantum Groups, Lecture Notes Math. 1510, (ed. Kulish, P.P.), Springer, 1992. 98–103.Google Scholar
19. Rahman, M., A simple proof of Koornwinder s addition formula for the little q-Legendre polynomials, Proc. Amer. Math. Soc. 107(1989), 373–381.Google Scholar
20. Rahman, M. and Verma, A., Product and addition formulas for the continuous q-ultraspherical polynomials, SIAM J. Math. Anal. 17(1986), 1461–1474.Google Scholar
21. Stanton, D., Orthogonal polynomials and Chevalley groups. In: Special Functions: Group Theoretical Aspects and Applications, (eds. Askey, R.A., Koornwinder, T.H. and Schempp, W.), Reidel, 1984. 87–128.Google Scholar
22. Van Assche, W. and Koornwinder, T.H., Asymptotic behaviour for Wall polynomials and the addition formula for little q-Legendre polynomials, SIAM J. Math. Anal. 22(1991), 302–311.Google Scholar