Lectures on q-Orthogonal Polynomials

  • Mourad E. H. Ismail
Part of the NATO Science Series book series (NAII, volume 30)

Abstract

These notes form a brief introduction to the theory of Askey-Wilson polynomials and related topics. We have used a novel approach to the development of those parts needed from the theory of basic hypergeometric functions. One important difference between our approach to basic hypergeometric functions and other approaches, for example those of Andrews, Askey, and Roy [7], of Gasper and Rahman [18] or of Bailey [12] is our frequent use of the Askey-Wilson divided difference operators, the q-difference operator, and the identity theorem for analytic functions.

We apply the bootstrap method from [13] and an attachment procedure to derive orthogonality relations for the Askey-Wilson polynomials in two steps using the Al-Salam-Chihara polynomials as an intermediate step. The continuous q-ultraspherical polynomialsand continuous q-Hermite polynomials are studied from a different approach. As an application of connection coefficient formulas we give the recent proof and generalizations of the Rogers-Ramanujan identities from [17].

These notes are summarized from a forthcoming Cambridge University Press monograph.

Keywords

Convolution Dick 

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References

  1. 1.
    N. I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis, English translation, Oliver and Boyed, Edinburgh, 1965.MATHGoogle Scholar
  2. 2.
    W. A. Al-Salam and T. S. Chihara, Convolutions of orthogonal polynomials, SIAM J. Math. Anal. 7 (1976), 16–28.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    G. E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Massachusetts, 1976.MATHGoogle Scholar
  4. 4.
    G. E. Andrews, q-series: Their development and application in analysis, number theory, combinatorics, physics, and computer algebra, CBMS Regional Conference Series, number 66, American Mathematical Society, Providence, R. I. 1986.Google Scholar
  5. 5.
    G. E. Andrews and R. A. Askey, A simple proof of Ramanujans summation 1ψ1, Aequationes Math. 18 (1978), 333–337.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    G. E. Andrews and R. A. Askey, Classical orthogonal polynomials, in: “Polynomes Orthogonaux et Applications”,C. Breziniski, et al., eds., Lecture Notes in Mathematics, Vol. 1171, Springer-Verlag, Berlin, 1984, pp. 36–63.Google Scholar
  7. 7.
    G. E. Andrews, R. A. Askey and R. Roy, Special Functions, Cambridge University Press, Cambridge, 1999.MATHGoogle Scholar
  8. 8.
    R. A. Askey, Divided difference operators and classical orthogonal polynomials, Rocky Mountain J. Math. 19 (1989), 33–37.MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    R. A. Askey and M. E. H. Ismail, A generalization of ultraspherical polynomials, in: “Studies in Pure Mathematics”, P. Erdös, ed., Birkhauser, Basel, 1983, pp. 55–78.Google Scholar
  10. 10.
    R. A. Askey and M. E. H. Ismail, Recurrence relations, continued fractions and orthogonal polynomials, Memoirs Amer. Math. Soc. Number 300 (1984).Google Scholar
  11. 11.
    R. A. Askey and J. A. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Memoirs Amer. Math. Soc. Number 319 (1985).Google Scholar
  12. 12.
    W. N. Bailey, Generalized Hypergeometric Series, Cambridge University Press, Cambridge, 1935.MATHGoogle Scholar
  13. 13.
    C. Berg and M. E. H. Ismail, q-Hermite polynomials and classical orthogonal polynomials, Canad. J. Math. 9 (1996), 43–63.MathSciNetCrossRefGoogle Scholar
  14. 14.
    D. Bressoud, On partitions, orthogonal polynomials and the expansion of certain infinite products, Proc. London Math. Soc. 42 (1981), 478–500.MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York, 1953.Google Scholar
  16. 16.
    E. Feldheim, Sur les polynomes généraisés de Legendre, Izv. Akad. Nauk. SSSR Ser. Math. 5 (1941), 241–248.MathSciNetMATHGoogle Scholar
  17. 17.
    K. Garrett, M. E. H. Ismail, and D. Stanton, Variants of the Rogers-Ramanujan identities, Advances in Appl. Math. 23 (1999), 274–299.MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge University Press, Cambridge, 1990.MATHGoogle Scholar
  19. 19.
    M. E. H. Ismail, A simple proof of Ramanujan’s 1σ1 sum, Proc. Amer. Math. Soc. 63 (1977), 185–186.MathSciNetMATHGoogle Scholar
  20. 20.
    M. E. H. Ismail, The Askey-Wilson operator and summation theorems, in: “Mathematical Analysis, Wavelets, and Signal Processing,”, M. E. H. Ismail, M. Z. Nashed, A. Zayed and A. Ghaleb, eds., Contemporary Mathematics, Vol. 190, Amer. Math. Soc, Providence, 1995, pp. 171–178.CrossRefGoogle Scholar
  21. 21.
    M. E. H. Ismail, D. Stanton, and G. Viennot, The combinatorics of the q-Hermite polynomials and the Askey-Wilson integral, European J. Combinatorics 8 (1987), 379–392.MathSciNetMATHGoogle Scholar
  22. 22.
    M. E. H. Ismail and J. Wilson, Asymptotic and generating relations for the q-Jacobi and the 4ø3 polynomials, J. Approx. Theory 36 (1982), 43–54.MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    I. L. Lanzewizky, Über die orthogonalität der Fejer-Szegösehen polynome, C. R. (Dokl.) Acad. Sci. URSS 31 (1941), 199–200.MathSciNetGoogle Scholar
  24. 24.
    A. P. Magnus, Associated Askey-Wilson polynomials as Laguerre-Hahn orthogonal polynomials, in: “Orthogonal Polynomials and Their Applications”, M. Alfaro et al., eds., Lecture Notes in Mathematics, Vol. 1329, Springer-Verlag, Berlin, 1988, pp. 261–278.CrossRefGoogle Scholar
  25. 25.
    M. Rahman and S. K. Suslov, Unified approach to the summation and integration formulas for q-hypergeometric functions I, J. Statistical Planning and Inference 54 (1996), 101–118.MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    G. Szegö, Orthogonal Polynomials, fourth edition, Amer. Math. Soc, Providence, 1975.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • Mourad E. H. Ismail
    • 1
  1. 1.Department of MathematicsUniversity of South FloridaTampaUSA

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