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Classical biorthogonal rational functions

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Methods of Approximation Theory in Complex Analysis and Mathematical Physics

Part of the book series: Lecture Notes in Mathematics ((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.

This work, supported in part by NSERC grant #A6197, was completed while the second author (SKS) was visiting Carleton University, Jan.–April, 1991

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References

  1. G.E. Andrews and R. Askey, Classical orthogonal polynomials, Polynomes orthogonaux et applications, Springer-Verlag, Berlin, Heidelberg, New York, 1985, pp. 36–62.

    MATH  Google Scholar 

  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. 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.

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. 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. 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. T.S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978.

    MATH  Google Scholar 

  9. G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge University Press, 1990.

    Google Scholar 

  10. W. Hahn, Über Orthogonal Polynome, die q-Differenzengleichungen genügen, Math. Nachr. 2 (1949), pp. 4–34.

    Article  MathSciNet  Google Scholar 

  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.

    MATH  Google Scholar 

  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. M. Rahman, Product and addition theorems for Hahn polynomials (to appear).

    Google Scholar 

  14. M. Rahman, Families of biorthogonal rational functions in a discrete variable, SIAM J. Math. Anal. 12 (1981), pp. 355–367.

    Article  MathSciNet  MATH  Google Scholar 

  15. M. Rahman, An integral representation of a 10φ9 and continuous bi-orthogonal 10φ9 rational functions, Canad. J. Nath. 38 (1986), pp. 605–618.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  18. M. Rahman and S.K. Suslov, The Pearson equation and the beta integrals, submitted.

    Google Scholar 

  19. S.K. Suslov, Classical orthogonality polynomials of a discrete variable, continuous orthogonality relations, Lett. Math. Phys. 14 (1987), pp. 77–88.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  21. J.A. Wilson, Some hypergeometric orthogonal polynomials, SIAM J. Math. Anal. 11 (1980), pp. 690–701.

    Article  MathSciNet  MATH  Google Scholar 

  22. J.A. Wilson, Orthogonal functions from Gram determinants, SIAM J. Math. Anal. 22 (1991), pp. 1147–1155.

    Article  MathSciNet  MATH  Google Scholar 

  23. J.A. Wilson, Hypergeometric series recurrence relations some new orthogonal functions, Ph.D. Thesis (1978), University of Wilsconsin, Madison, Wisc..

    Google Scholar 

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Author information

Authors and Affiliations

  1. Dept. of Mathematics and Statistics, Carleton University, K1S 5B6, Ottawa, Ontario, Canada

    Mizan Rahman

  2. Kurchatov Institute of Atomic Energy, 123182, Moscow, Russia

    S. K. Suslov

Authors
  1. Mizan Rahman
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  2. S. K. Suslov
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Editor information

Andrei A. Gonchar Edward B. Saff

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© 1993 The Euler International Mathematical Institute

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Rahman, M., Suslov, S.K. (1993). Classical biorthogonal rational functions. In: Gonchar, A.A., Saff, E.B. (eds) Methods of Approximation Theory in Complex Analysis and Mathematical Physics. Lecture Notes in Mathematics, vol 1550. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0117478

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  • DOI: https://doi.org/10.1007/BFb0117478

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  • Print ISBN: 978-3-540-56931-2

  • Online ISBN: 978-3-540-47792-1

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