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Bounds for the Extreme Zeros of Laguerre Polynomials

  • Geno Nikolov
  • Rumen UluchevEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11189)

Abstract

By applying well-known techniques such as the Gershgorin Circle Theorem and the Euler-Rayleigh method (the latter assisted by some computer algebra), we obtain new bounds for the extreme zeroes of the n-th Laguerre polynomial. It turns out that these bounds are competitive to some of the known best bounds.

Keywords

Extreme zeros of Laguerre polynomials Gershgorin circle theorem Euler-Rayleigh method 

References

  1. 1.
    Bottema, Q.: Die Nullstellen gewisser durch Rekursionsformeln definierter Polynome. Proc. Amsterdam 34(5), 681–691 (1931)zbMATHGoogle Scholar
  2. 2.
    Chihara, T.: An Introduction to Orthogonal Polynomials. Gorn and Breach, New York (1978)zbMATHGoogle Scholar
  3. 3.
    Dimitrov, D.K., Nikolov, G.P.: Sharp bounds for the extreme zeros of classical orthogonal polynomials. J. Approx. Theory 162, 1793–1804 (2010)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Driver, K., Jordaan, K.: Bounds for extreme zeros of some classical orthogonal polynomials. J. Approx. Theory 164, 1200–1204 (2012)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Driver, K., Jordaan, K.: Inequalities for extreme zeros of some classical orthogonal and \(q\)-orthogonal polynomials. Math. Model. Nat. Phenom. 8(1), 48–59 (2013)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Gupta, D.P., Muldoon, M.E.: Inequalities for the smallest zeros of Laguerre polynomials and their \(q\)-analogues. J. Ineq. Pure Appl. Math. 8(1) (2007). Article 24Google Scholar
  7. 7.
    Hahn, W.: Bericht über die Nullstellen der Laguerreschen und der Hermiteschen Polynome. Jahresber. Deutsch. Math.-Ferein. 44, 215–236 (1933)zbMATHGoogle Scholar
  8. 8.
    Ismail, M.E.H., Muldoon, M.E.: Bounds for the small real and purelyimaginary zeros of Bessel and related functions. Met. Appl. Math. Appl. 2, 1–21 (1995)zbMATHGoogle Scholar
  9. 9.
    Ismail, M.E.H., Li, X.: Bounds on the extreme zeros of orthogonal polynomials. Proc. Amer. Math. Soc. 115, 131–140 (1992)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Krasikov, I.: Bounds for zeros of the Laguerre polynomials. J. Approx. Theory 121, 287–291 (2003)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Neumann, E.R.: Beiträge zur Kenntnis der Laguerreschen Polynome. Jahresber. Deutsch. Math.-Ferein 30, 15–35 (1921)zbMATHGoogle Scholar
  12. 12.
    Szegő, G.: Orthogonal Polynomials, 4th edn. American Mathematical Society Colloquium Publications, Providence (1975)zbMATHGoogle Scholar
  13. 13.
    Van der Waerden, B.L.: Modern Algebra, vol. 1. Frederick Ungar Publishing Co., New York (1949)Google Scholar
  14. 14.
    van Dorn, E.: Representations and bounds for zeros of orthogonal polynomials and eigenvalues of sign-symmetric tri-diagonal matrices. J. Approx. Theory 51, 254–266 (1987)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Faculty of Mathematics and InformaticsSofia University “St. Kliment Ohridski”SofiaBulgaria

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