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Interlacing of zeros of shifted sequences of one-parameter orthogonal polynomials

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Abstract

We study the interlacing property of zeros of Laguerre polynomials of adjacent degree, where the free parameters differ by an integer, and of the same degree, where the free parameter is shifted continuously. Similar interlacing results are proven for the positive zeros of Gegenbauer polynomials.

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References

  1. Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Nat. Bur. Standards Appl. Series, vol. 55, US Government Printing Office, Washington, DC (1964)

  2. Andrews, G.E., Askey R.: Classical orthogonal polynomials. In: Brezinski, C., et al. (eds.) Polynômes Orthogonaux et Applications. Lecture Notes in Mathematics, 1171, pp. 36–62. Springer, New York (1985)

  3. Andrews, G., Askey, R., Roy, R.: Special Functions. Encycl. of Math and its Applic., Cambridge University Press (1999)

  4. Beardon A.F., Driver K.A. (2005). The zeros of linear combinations of orthogonal polynomials. J. Approx. Theory 137: 179–186

    Article  MATH  MathSciNet  Google Scholar 

  5. Brezinski C., Driver K.A., Redivo-Zaglia M. (2004). Quasi-orthogonality with applications to some families of classical orthogonal polynomials. Appl. Numer. Math. 48(2): 157–168

    Article  MATH  MathSciNet  Google Scholar 

  6. Criscuolo G., Mastroianni G., Occorsio D. (1990). Convergence of extended Lagrange interpolation. Math. Comp. 55n(191): 197–212

    Article  MathSciNet  Google Scholar 

  7. Criscuolo G., Mastroianni G., Occorsio D. (1991). Uniform convergence of derivatives of extended Lagrange interpolation. Numer. Math. 60: 195–218

    Article  MATH  MathSciNet  Google Scholar 

  8. Driver K., Jordaan K. (2007). Separation theorems for the zeros of certain hypergeometric polynomials. J. Comput. Appl. Math. 199(1): 48–55

    Article  MATH  MathSciNet  Google Scholar 

  9. Joulak H. (2005). A contribution to quasi-orthogonal polynomials and associated polynomials. Appl. Numer. Math. 54: 65–78

    Article  MATH  MathSciNet  Google Scholar 

  10. Levit R.J. (1967). The zeros of the Hahn polynomials. SIAM Rev. 9(2): 191–203

    Article  MATH  MathSciNet  Google Scholar 

  11. Locher, F.: Stability test for linear difference forms, Numerical integration, IV (Oberwolfach,1992), pp. 215–223, Internat. Ser. Numer. Math. 112 (1993)

  12. Mastroianni G., Occorsio D. (1995). Interlacing properties of the zeros of the orthogonal polynomials and approximation of the Hilbert transform. Comput. Math. Appl. 30(3–6): 155–168

    Article  MATH  MathSciNet  Google Scholar 

  13. Mastroianni G., Occorsio D. (2001). Optimal systems of nodes for Lagrange Interpolation on bounded intervals. A survey. J. Comput. Appl. Math. 134: 325–341

    Article  MATH  MathSciNet  Google Scholar 

  14. Occorsio D. (1994). Convergence of extended Lagrange interpolation in weighted L p norm. Calcolo 31: 47–61

    Article  MATH  MathSciNet  Google Scholar 

  15. Peherstorfer F. (1990). Linear combination of orthogonal polynomials generating positive quadrature formulas. Math. Comput. 55(191): 231–241

    Article  MATH  MathSciNet  Google Scholar 

  16. Szeg̋ G. (1959). Orthogonal Polynomials. American Mathematical Society, New York

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag X3, Rondebosch, 7701, Cape Town, South Africa

    Kathy Driver

  2. Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, 0002, South Africa

    Kerstin Jordaan

Authors
  1. Kathy Driver
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  2. Kerstin Jordaan
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Correspondence to Kerstin Jordaan.

Additional information

Research by Kathy Driver is supported by the National Research Foundation of South Africa under grant number 2053730.

Research by Kerstin Jordaan is partially supported by the National Research Foundation of South Africa under grant number 2054423.

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Driver, K., Jordaan, K. Interlacing of zeros of shifted sequences of one-parameter orthogonal polynomials. Numer. Math. 107, 615–624 (2007). https://doi.org/10.1007/s00211-007-0100-3

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  • DOI: https://doi.org/10.1007/s00211-007-0100-3

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