Advertisement

Classical biorthogonal rational functions

  • Mizan Rahman
  • S. K. Suslov
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1550)

Abstract

A general set of biorthogonal rational functions, considered previously by Rahman and Wilson, is shown to satisfy a second-order linear difference equation of a nonuniform lattice. In the spirit of Hahn’s approach for orthogonal polynomials, raising and lowering operators as well as a Rodriguez-type formula are obtained for these functions which contain the classical orthogonal polynomials as limiting cases. Their biorthogonality in the discrete case is established by means of a Sturm-Liouville type argument. An outline of Wilson’s technique for representing them as Gram determinants is also given.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    G.E. Andrews and R. Askey, Classical orthogonal polynomials, Polynomes orthogonaux et applications, Springer-Verlag, Berlin, Heidelberg, New York, 1985, pp. 36–62.zbMATHGoogle Scholar
  2. [2]
    R. Askey, Beta integrals and q-extensions, Proc. Ramanujan Centennial International Conference (eds. R. Balakrishnan, K.S. Padmanabhan and V. Thangaraj), Ramanujan Math. Soc., Annamalai University, Annamalainagar, 1988, pp. pp. 85–102.Google Scholar
  3. [3]
    R. Askey and J.A. Wilson, A set of orthogonal polynomials that generalize the Racah coefficients or 6-j symbols, SIAM J. Math. Anal. 10 (1979), pp. 1008–1016.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    R. Askey and J.A. Wilson, Some basic hypergeometric polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. no. #319 (1985).Google Scholar
  5. [5]
    N.M. Atakishiyev. G.I. Kuznetsov and S.K. Suslov, A definition and a classification of the classical orthogonal polynomials, in preparation.Google Scholar
  6. [6]
    N.M. Atakishiyev. G.I. Kuznetsov and S.K. Suslov, The Charm polynomials, Proceedings of the Third International Symposium on Orthogonal Polynomials and Their Applications, (Berzinski, Gori and Ronveaux, eds.) J.C. Baltzer AG, Basel, Switzerland, 1991, pp. 15–16.Google Scholar
  7. [7]
    P.L. Chebyshev, Complete collected works in 5 volumes, Izdat. Akad. Nauk. SSSR, Moscow, 1947–1957; On interpolation of values of equidistants, 1975, pp. 66–87. (in Russian)Google Scholar
  8. [8]
    T.S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978.zbMATHGoogle Scholar
  9. [9]
    G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge University Press, 1990.Google Scholar
  10. [10]
    W. Hahn, Über Orthogonal Polynome, die q-Differenzengleichungen genügen, Math. Nachr. 2 (1949), pp. 4–34.MathSciNetCrossRefGoogle Scholar
  11. [11]
    A.F. Nikiforov, S.K. Suslov and V.B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable, Nauka, Moscow, 1985 (in Russian); English translation, Springer-Verlag.zbMATHGoogle Scholar
  12. [12]
    A.F. Nikiforov and S.K. Suslov, Lett. Math. Phys. 11 no. 1 (1986), pp. 27–34, Classical orthogonal polynomials of a discrete variable on nonuniform lattices.Google Scholar
  13. [13]
    M. Rahman, Product and addition theorems for Hahn polynomials (to appear).Google Scholar
  14. [14]
    M. Rahman, Families of biorthogonal rational functions in a discrete variable, SIAM J. Math. Anal. 12 (1981), pp. 355–367.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    M. Rahman, An integral representation of a 19 and continuous bi-orthogonal 19 rational functions, Canad. J. Nath. 38 (1986), pp. 605–618.MathSciNetCrossRefGoogle Scholar
  16. [16]
    M. Rahman, Biorthogonality of a system of rational functions with respect to a positive measure on [−1, 1], SIAM J. Math. Anal. 22 (1991), pp. 1430–1441.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    M. Rahman, Some extensions of Askey-Wilson's q-beta integral and the corresponding orthogonal systems, Canad. Math. Bull. 31 (1988), pp. 467–476.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    M. Rahman and S.K. Suslov, The Pearson equation and the beta integrals, submitted.Google Scholar
  19. [19]
    S.K. Suslov, Classical orthogonality polynomials of a discrete variable, continuous orthogonality relations, Lett. Math. Phys. 14 (1987), pp. 77–88.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    S.K. Suslov, The theory of difference analogues of special functions of hypergeometric type, Russian Math. Surveys 44 no. 2 (1989), pp. 227–278.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    J.A. Wilson, Some hypergeometric orthogonal polynomials, SIAM J. Math. Anal. 11 (1980), pp. 690–701.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    J.A. Wilson, Orthogonal functions from Gram determinants, SIAM J. Math. Anal. 22 (1991), pp. 1147–1155.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    J.A. Wilson, Hypergeometric series recurrence relations some new orthogonal functions, Ph.D. Thesis (1978), University of Wilsconsin, Madison, Wisc..Google Scholar

Copyright information

© The Euler International Mathematical Institute 1993

Authors and Affiliations

  • Mizan Rahman
    • 1
  • S. K. Suslov
    • 2
  1. 1.Dept. of Mathematics and StatisticsCarleton UniversityOttawaCanada
  2. 2.Kurchatov Institute of Atomic EnergyMoscowRussia

Personalised recommendations