Constructive Approximation

, Volume 7, Issue 1, pp 521–534 | Cite as

Associated Wilson polynomials

  • David R. Masson
Article

Abstract

From a contiguous relation obtained by Wilson for terminating 2-balanced very well-poised9F8 hypergeometric functions of unit argument, we derive a pair of three term recurrence relations for very well-poised7F6's. From these we obtain solutions to the recurrence relation for associated Wilson polynomials and spectral properties of the corresponding Jacobi matrix. A calculation of the basic weight function yields a generalization of Dougall's theorem.

AMS classification

33C20 33C45 30B70 40A15 39A10 

Key words and phrases

Orthogonal polynomials Hypergeometric functions Contiguous relations Three term recurrence relations Jacobi matrix Spectral properties Dougall's theorem 

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References

  1. 1.
    N. I. Akhiezer (1965): The Classical Moment Problem. Edinburgh: Oliver and Boyd.Google Scholar
  2. 2.
    R.Askey, J.Wilson (1985):Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Memoirs Amer. Math. Soc.,319.Google Scholar
  3. 3.
    R. Askey, J. Wimp (1984):Associated Laguerre and Hermite polynomials. Proc. Royal Soc. Edinburgh,96A: 15–37.Google Scholar
  4. 4.
    W. N. Bailey (1935): Generalized Hypergeometric Series, Tract No. 32. (Cambridge: Cambridge University Press).Google Scholar
  5. 5.
    W. Gautschi (1967):Computational aspects of three-term recurrence relations. SIAM Review,9: 24–82.Google Scholar
  6. 6.
    M. E. H. Ismail, J. Letessier, G. Valent (1988):Linear birth and death models and associated Laguerre polynomials. J. Approx. Theory,56: 337–348.Google Scholar
  7. 7.
    M. E. H. Ismail, J. Letessier, G. Valent (1989):Quadratic birth and death processes and associated continuous dual Hahn polynomials. SIAM J. Math. Anal.,20: 727–737.Google Scholar
  8. 8.
    M. E. H. Ismail, J. Letessier, G. Valent, J. Wimp (1990):Two families of associated Wilson polynomials. Canadian J. Math.,42:659–695.Google Scholar
  9. 9.
    M. E. H.Ismail, D. R.Masson (to appear):Two families of orthogonal polynomials related to Jacobi polynomials. Rocky Mountain J. Math.Google Scholar
  10. 10.
    W. B. Jones, W. J. Thron (1980): Continued Fractions, Analytic Theory and Applications. Reading, MA: Addison-Wesley.Google Scholar
  11. 11.
    D. Masson (1983):The rotating harmonic oscillator eigenvalue problem, I. Continued fractions and analytic continuation. J. Math. Phys.,24: 2074–2088.Google Scholar
  12. 12.
    D. R. Masson (1988):Difference equations, continued fractions, Jacobi matrices and orthogonal polynomials. In: Nonlinear Numerical Methods and Rational Approximation (A. Cuyt, ed.). Dordrecht: Reidel, pp. 239–257.Google Scholar
  13. 13.
    D. R.Masson (to appear):Wilson polynomials and some continued fractions of Ramanujan. Rocky Mountain J. Math.Google Scholar
  14. 14.
    J. Raynal (1979):On the definition and properties of generalized 6 — j symbols. J. Math. Phys.,20: 2398–2415.Google Scholar
  15. 15.
    F. W. J. Whipple (1936):Relations between well-poised hypergeometric series of type 7F6. London Math. Soc., (2)40: 336–344.Google Scholar
  16. 16.
    J. A. Wilson (1977):Three-term contiguous relations and some new orthogonal polynomials. In: Padé and Rational Approximation, Proc. Internat. Sympos., Univ. of South Florida, Tampa, FL., 1976. New York: Academic Press, pp. 227–232.Google Scholar
  17. 17.
    J. A.Wilson (1978): Hypergeometric Series Recurrence Relations and Some New Orthogonal Functions (Thesis, University of Wisconsin-Madison).Google Scholar
  18. 18.
    J. A. Wilson (1980):Some hypergeometric orthogonal polynomials. SIAM J. Math. Anal,11: 690–701.Google Scholar
  19. 19.
    J. Wimp (1987):Explicit formulas for the associated Jacobi polynomials and some applications. Canadian J. Math.,39: 983–1000.Google Scholar

Copyright information

© Springer-Verlag New York Inc 1991

Authors and Affiliations

  • David R. Masson
    • 1
  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada

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