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The distribution of zeros of the polynomial eigenfunctions of ordinary differential operators of arbitrary order

  • E. Buendia
  • J. S. Dehesa
  • F. J. Galvez
Contributors
Part of the Lecture Notes in Mathematics book series (LNM, volume 1329)

Abstract

The distribution of zeros of the polynomial eigenfunctions of ordinary differential operators of arbitrary order with polynomial coefficients is calculated via its moments directly in terms of the parameters which characterize the operators. Some results of K.M. Case and the authors are extended. In particular, the restriction for the degree of the polynomial coefficient of the ith-derivative to be not greater than i is relaxed. Applications to the Heine polynomials, the Generalized Hermite polynomials and the Bessel type orthogonal polynomials (which no trivial properties of zeros were known of) are shown.

Keywords

Orthogonal Polynomial Arbitrary Order Ordinary Differential Equation Polynomial Coefficient Polynomial Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    H.L. KRALL, Certain Differential Equations for Tchebycheff Polynomials, Duke Math. J. 4(1938),705–718.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    H.L. KRALL, On Orthogonal Polynomials Satisfying a Certain Fourth Order Differential Equation", Pennsylvania State College Studies, No.6, Pennsylvania St. College, State College. (1940).Google Scholar
  3. 3.
    A.M. KRALL, Tchebycheff Sets of Polynomials which Satisfy an Ordinary Differential Equation, SIAM Rev. 22(1980),436–441MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    A.M. KRALL, Orthogonal Polynomials Satisfying Fourth Order Differential Equations, Proc. Roy. Soc. Edinburgh, A87 (1981),271–288MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    A.M. KRALL, A Review of Orthogonal Polynomials Satisfying Boundary Value Problems, Contribution to this volume.Google Scholar
  6. 6.
    L.L. LITTLEJOHN, On the Classification of Differential Equations Having Orthogonal Polynomial Solutions. I, Annali di Matem. 138 (1984), 35–53. II, to appear in Annali di Matem. (1986).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    L.L. LITTLEJOHN, Orthogonal Polynomials Solution to Ordinary Partial Differential Equations, Contribution to this volume.Google Scholar
  8. 8.
    W. HAHN, On Differential Equation For Orthogonal Polynomials, Funkcialaj Ekvacioj 21(1978),1–9MathSciNetzbMATHGoogle Scholar
  9. 9.
    F.V. ATKINSON and W.N. EVERITT, Orthogonal Polynomials Satisfying Second Order Differential Equations, Christoffel Festschrift (1981) Birkhauser Verlag, Basel), p. 173–181Google Scholar
  10. 10.
    R. RASALA, The Rodrigues Formula and Polynomial Differential Operators, J. Math. Anal. Appl. 84(1981),443–482.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    R. SMITH, An Abundance of Orthogonal Polynomials, IMA J. Appl. Math. 28(1982)161–167.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    T.H. KOORNWINDER, Orthogonal Polynomials with Weight Function (1-x)α (1+x)β + ML(x+1) + NL(x-1). Canad. Math. Bull. 27 (1984),205–214MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    K.M. CASE, Sum Rules for Zeros of Polynomials. I,J. Math. Phys. 21(1980),702–708MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    J.S. DEHESA, E. BUENDIA and M.A. SANCHEZ BUENDIA, On the Polynomial Solutions of Ordinary Differential Equations of the Fourth Order, J. Math. Phys. 26(1985),1547–1552MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    E. BUENDIA, J.S. DEHESA and M.A. SANCHEZ BUENDIA, On the Zeros of Eigenfunctions of Polynomial Differential Operators,J. Math. Phys. 26(1985),2729–2736.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    K.M. CASE, Sum Rules for Zeros of Polynomials. II, J. Math. Phys. 21(1980),709–714MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    T.S. CHIHARA,An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1977zbMATHGoogle Scholar
  18. 18.
    G. SZEGO, Ein Beitrag zur Theorie der Polynome von Laguerre und Jacobi, Math. Zeit. 1(1918),341–356.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    E. HENDRIKSEN, A Bessel Type Orthogonal Polynomial System, Indagations Math. 46(1984),407–414MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    E. BUENDIA and J.S. DEHESA, A New Class of Sum Rules for Zeros of Polynomials, Preprint of Granada University, 1986.Google Scholar
  21. 21.
    J. RIORDAN, An Introduction to Combinatorial Analysis. Wiley, N. Y., 1958.zbMATHGoogle Scholar
  22. 22.
    N.E. HEINE, Handbuch der Kugelfunctionen, I. Reiman, Berlin, 1878.Google Scholar
  23. 23.
    K. AOMOTO, Lax Equation and the Spectral Density of Jacobi Matrices for Orthogonal Polynomials, Nagoya preprintGoogle Scholar
  24. 24.
    A.M. KRALL, On the Generalized Hermite Polynomials, Indiana Math. J. 30(1981)73–78.MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    F.J. GALVEZ and J.S. DEHESA, Some Open Problems of Generalized Bessel Polynomials, J. Phys. A: Math. Gen.17(1984),2759–2766.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • E. Buendia
    • 1
  • J. S. Dehesa
    • 1
  • F. J. Galvez
    • 1
  1. 1.Departamento de Fisica NuclearUniversidad de GranadaGranadaSpain

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