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Orthogonal Polynomials in Connection with Quantum Groups

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Orthogonal Polynomials

Part of the book series: NATO ASI Series ((ASIC,volume 294))

Abstract

This is a survey of interpretations of q-hypergeometric orthogonal polynomials on quantum groups. The first half of the paper gives general background on Hopf algebras and quantum groups. The emphasis in the rest of the paper is on the SU(2) quantum group. An interpretation of little q-Jacobi polynomials as matrix elements of its irreducible representations is presented. In the last two sections new results by the author on interpretations of Askey-Wilson polynomials are discussed.

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References

  1. E. Abe, Hopf Algebras, Cambridge University Press, 1980.

    Google Scholar 

  2. G. E. Andrews an R. Askey, ‘Enumeration of partitions: the role of Eulerian series and q-orthogonal polynomials’, pp. 3-26 in Higher Combinatorics(M. Aigner, ed. ), Reidel, 1977.

    Google Scholar 

  3. M. Arik and D. D. Coon, ‘Hilbert spaces of analytic functions and generalized coherent states’, J. Math. Phys. 17(1976), 524 – 527.

    Article  MathSciNet  Google Scholar 

  4. R. Askey and M. E. H. Ismail, ‘A generalization of ultrasphericai polynomials’, pp. 55–78 in Studies in Pure Mathematics(P. Erdös, ed. ), Birkhäuser, 1983.

    Google Scholar 

  5. R. Askey and J. Wilson, ‘A set of orthogonal polynomials that generalize the Racah coefficients or 6-jsymbols’, SIAM J. Math. Anal. 10(1979), 1008 – 1016.

    Article  MathSciNet  MATH  Google Scholar 

  6. R. Askey and J. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Memoirs Amer. Math. Soc. 54 (1985) No. 319.

    Google Scholar 

  7. W. Van Assche and T. H. Koornwinder, Asymptotic behaviour for Wall polynomials and the addition formula for little q-Legendre polynomials, preprint, 1989.

    Google Scholar 

  8. G. M. Bergman, ‘Everybody knows what a Hopf algebra is’, Contemp. Math. 43(1985), 25 – 48.

    MathSciNet  Google Scholar 

  9. P. Cartier, Harmonic analysis on trees, Proc. Sympos. Pure Math. 26(1973), 419 – 424.

    Google Scholar 

  10. J. Cigler, ‘Operatormethoden für q-Identitäten’, Monatsh. Math. 88(1979), 87 – 105.

    Article  MathSciNet  MATH  Google Scholar 

  11. V. G. Drinfeld, ‘Quantum groups’, pp. 798–820 in Proceedings of the International Congress of Mathematicians, Berkeley, 1986, American Mathematical Society, 1987.

    Google Scholar 

  12. Ph. Feinsilver, ‘Commutators, anti-commutators and Eulerian calculus’, Rocky Mountain J. Math. 12(1982), 171 – 183.

    Article  MathSciNet  MATH  Google Scholar 

  13. Ph. Feinsilver, ‘Discrete analogues of the Heisenberg-Weyl algebra’, Monatsh. Math. 104(1987), 89 – 108.

    Article  MathSciNet  MATH  Google Scholar 

  14. G. Gasper and M. Rahman, Basic hypergeometric series, Cambridge University Press, 1989.

    Google Scholar 

  15. M. Hazewinkel, Formal groups and applications, Academic Press, 1978.

    Google Scholar 

  16. M. Jimbo, ‘A q-difference analogue of U(g)and the Yang-Baxter equation’, Lett. Math. Phys. 10(1985), 63 – 69.

    Article  MathSciNet  MATH  Google Scholar 

  17. A. N. Kirillov and N. Yu. Reshetikhin, Representations of the algebra U q (sl(2)), q- orthogonal polynomials and invariants of links, LOMI Preprints E-9-88, Leningrad, 1988.

    Google Scholar 

  18. H. T. Koelink and T. H. Koornwinder, ’The Clebsch-Gordan coefficients for the quantum group S μ U(2) and q-Hahn polynomials’, Nederl. Akad. Wetensch. Proc. Ser. A, to appear.

    Google Scholar 

  19. T. H. Koornwinder, ‘Clebsch-Gordan coefficients for SU(2) and Hahn polynomials’, Nieuw Archief Wisk. (3) 29(1981), 140 – 155.

    MathSciNet  MATH  Google Scholar 

  20. T. H. Koornwinder, ‘Krawtchouk polynomials, a unification of two different group theoretic interpretations’, SIAM J. Math. Anal. 13(1982), 1011 – 1023.

    Article  MathSciNet  MATH  Google Scholar 

  21. T. H. Koornwinder, ‘Representations of the twisted SU(2) quantum group and some q-hypergeometric orthogonal polynomials’, Nederl. Akad. Wetensch. Proc. Ser. A 92(1989), 97 – 117.

    MathSciNet  Google Scholar 

  22. T. H. Koornwinder, The addition formula for little q-Legendre polynomials and the SU(2) quantum group, CWI Rep. AM-R8906, preprint, 1989.

    Google Scholar 

  23. T. H. Koornwinder, ‘Continuous q-Legendre polynomials are spherical matrix elements of irreducible representations of the quantum SU(2) group’, CWI Quarterly to appear.

    Google Scholar 

  24. I. G. Macdonald, Orthogonal polynomials associated with root systems, preprint, 1988.

    Google Scholar 

  25. I. G. Macdonald, ‘Orthogonal polynomials associated with root systems’, These Proceedings

    Google Scholar 

  26. Yu. I. Manin, Quantum groups and non-commutative geometry, Centre de Recherches Mathématiques, Montréal, 1988.

    Google Scholar 

  27. T. Masuda, K. Mimachi, Y. Nakagami, M. Noumi and K. Ueno, ‘Representations of quantum groups and a q-analogue of orthogonal polynomials’, C. R. Acad. Sci. Paris, Sér. I Math. 307(1988), 559 – 564.

    MathSciNet  MATH  Google Scholar 

  28. T. Masuda, K. Mimachi, Y. Nakagami, M. Noumi and K. Ueno, ‘Representations of the quantum group SUq(2) and the little q-Jacobi polynomials’J. Functional Anal to appear.

    Google Scholar 

  29. T. Masuda, K. Mimachi, Y. Nakagami, M. Noumi, Y. Saburi and K. Ueno, Unitary representations of the quantum group SU q(1,1), I, II, preprint, 1989.

    Google Scholar 

  30. A. Nijenhuis and R. W. Richardson, ‘Deformations of Lie algebra structures’, J. Math. Mech. 17(1967), 89 – 105.

    MathSciNet  MATH  Google Scholar 

  31. A. F. Nikiforov and V. B. Ugarov, Special Functions of Mathematical Physics, Birkhäuser, 1988.

    Google Scholar 

  32. M. Noumi and K. Mimachi, Quantum 2-spheres and big q-Jacobipolynomials, preprint, 1989.

    Google Scholar 

  33. M. Noumi and K. Mimachi, Big q-Jacobipolynomials, q-Hahn polynomials and a family of quantum 3-spheres, preprint, 1989.

    Google Scholar 

  34. M. Noumi, H. Yamada and K. Mimachi, Zonal spherical functions on the quantum homogeneous space SU q(n +1)/SU q(n), Proc. Japan Acad. 65(1989), 169 – 171.

    Article  MathSciNet  MATH  Google Scholar 

  35. P. Podleś, ‘Quantum spheres’, Lett. Math. Phys. 14(1987), 193 – 202.

    Article  MathSciNet  MATH  Google Scholar 

  36. M. Rahman, A simple proof ofKoornwinder’s addition formula for the little q-Legendre polynomials, preprint, 1988.

    Google Scholar 

  37. M. Rahman, ‘Some extensions of the beta integral and the hypergeometrie function’, These Proceedings

    Google Scholar 

  38. M. Rahman and A. Verma, ‘Product and addition formula for the continuous q-ultra- spherical polynomials’, SIAM J. Math. Anal. 17(1986), 1461 – 1474.

    Article  MathSciNet  MATH  Google Scholar 

  39. M. P. Schützenberger, ‘Une interprétation de certaines solutions de l’équation fonc- tionnelle: F(x + y) = F(x)F(y)’, C. R. Acad. Sci. Paris 236(1953), 352 – 353.

    Google Scholar 

  40. D. Stanton, ‘Orthogonal polynomials and Chevalley groups’ pp. 87–128 in Special Functions: Group Theoretic Aspects and Applications(R. A. Askey, T. H. Koornwinder and W. Schempp, eds. ), Reidel, 1984.

    Google Scholar 

  41. D. Stanton, ‘An introduction to group representations and orthogonal polynomials’, These Proceedings

    Google Scholar 

  42. M. E. Sweedler, Hopf Algebras, Benjamin, 1969

    Google Scholar 

  43. L. L. Vaksman, q-Analogues of Clebsch-Gordan coefficients in the algebra of functions on the quantum group SU(2), 1989.

    Google Scholar 

  44. L. L. Vaksman and L. I. Korogodsky, ‘Algebra of bounded functions on the quantum group of plane motions and q-analogues of Bessel functions’, Dokl. Akad. Nauk SSSR 304 (1989), 1036–1040 (in Russian).

    Google Scholar 

  45. L. L. Vaksman and Ya. S. Soibelman, ‘Algebra of functions on the quantum group SU(2)’, Functional Anal. Appl. 22(1988), 170 – 181.

    Article  MathSciNet  MATH  Google Scholar 

  46. J.-M. Vallin, ’C*-algèbres de Hopf et C*-algèbres de Kac’, Proc. London Math. Soc. (3) 50(1985), 131 – 174.

    Article  MathSciNet  MATH  Google Scholar 

  47. N. Ya. Vilenkin, Special Functions and the Theory of Group Representations, Amer. Math. Soc. Transl. of Math. Monographs, Vol. 22, 1968.

    Google Scholar 

  48. J. A. Wilson, ‘Some hypergeometric orthogonal polynomials’, SIAM J. Math. Anal. 11(1980), 690 – 701.

    Article  MathSciNet  MATH  Google Scholar 

  49. S. L. Woronowicz, ‘Compact matrix pseudogroups’, Comm. Math. Phys. 111(1987), 613 – 665.

    Article  MathSciNet  MATH  Google Scholar 

  50. S. L. Woronowicz, ‘Twisted SU(2) group. An example of a non-commutative differential calculus’, Publ. Res. Inst. Math. Sci. 23(1987), 117 – 181.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Centre for Mathematics and Computer Science, P.O. Box 4079, 1009 AB, Amsterdam, The Netherlands

    Tom H. Koornwinder

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  1. Tom H. Koornwinder
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Editors and Affiliations

  1. Department of Mathematics, The Ohio State University, Columbus, Ohio, USA

    Paul Nevai

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© 1990 Kluwer Academic Publishers

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Koornwinder, T.H. (1990). Orthogonal Polynomials in Connection with Quantum Groups. In: Nevai, P. (eds) Orthogonal Polynomials. NATO ASI Series, vol 294. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-0501-6_12

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  • DOI: https://doi.org/10.1007/978-94-009-0501-6_12

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-6711-9

  • Online ISBN: 978-94-009-0501-6

  • eBook Packages: Springer Book Archive

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