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Orthogonal Polynomials with Respect to Modified Jacobi Weight and Corresponding Quadrature Rules of Gaussian Type

Published online by Cambridge University Press:  28 May 2015

Marija P. Stanić  and
Aleksandar S. Cvetković
Marija P. Stanić*
Affiliation:
Department of Mathematics and Informatics, Faculty of Science, University of Kragujevac, Radoja Domanovi´ca 12, 34000 Kragujevac, Serbia
Aleksandar S. Cvetković*
Affiliation:
Department of Mathematics and Informatics, Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120 Belgrade, Serbia
*
Corresponding author.Email address:stanicm@kg.ac.rs
Corresponding author.Email address:acvetkovic@mas.bg.ac.rs
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Abstract

In this paper we consider polynomials orthogonal with respect to the linear functional defined on the space of all algebraic polynomials by

where α,β> -1/2 are real numbers such that ℓ = |β - α| is a positive integer, and ζ∈ R{0}. We prove the existence of such orthogonal polynomials for some pairs of α and ζ and for all nonnegative integers ℓ. For such orthogonal polynomials we derive three-term recurrence relations and also some differential-difference relations. For such orthogonal polynomials the corresponding quadrature rules of Gaussian type are considered. Also, some numerical examples are included.

Keywords

33C4741A5565D30Orthogonal polynomialsmodified Jacobi weight functionrecurrence relationGaussian quadrature rule
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 4 , Issue 4 , November 2011 , pp. 478 - 488
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]Chihara, T. S., An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978.Google Scholar
[2]Cvetković, A. S. and Milovanović, G. V., The Mathematica package “OrthogonalPolynomials”, FactaUniv. Ser. Math. Inform., 19, (2004), pp. 17–36.Google Scholar
[3]Gautschi, W. and Milovanović, G. V., Polynomials orthogonal on the semicircle, J. Approx. Theory, 46, (1986) pp. 230–250.CrossRefGoogle Scholar
[4]Gradshteyn, I. S. and Ryzhik, I. M., Table of Integrals, Series, and Products, sixth edition, Academic Press, London, 2000.Google Scholar
[5]Milovanović, G.V. and Cvetković, A. S., Orthogonal polynomials and Gaussian quadrature rules related to oscillatory weight functions, J. Comput. Appl. Math., 179, (2005), pp. 263–287.CrossRefGoogle Scholar
[6]Milovanović, G.V. and Cvetković, A. S., Orthogonal polynomials related to the oscillatory- Chebyshev weight function, Bull. Cl. Sci. Math. Nat. Sci. Math., 30, (2005), pp. 47–60.Google Scholar
[7]Milovanović, G.V., Cvetković, A. S., and Marjanović, Z. M., Orthogonal polynomials for the oscillatory-Gegenbauer weight, Publ. Inst. Math. (Beograd) (N.S.), 84(98), (2008), pp. 49–60.CrossRefGoogle Scholar
[8]Milovanović, G.V., Cvetković, A. S., and Stanić, M. P., Orthogonal polynomials for modified Gegenbauer weight and corresponding quadratures, Appl. Math. Lett., 22(8), (2009), pp. 1189–1194.CrossRefGoogle Scholar
[9]Orfanidis, S. J., Electromagnetic Waves and Antennas, Rutgers University, NJ, 2008, Google Scholar
[10]Watson, G. N., A Treatise on the Theory ofBessel Functions, Cambridge Univ. Press, 1922.Google Scholar
[11]Šidlovskiĭ, A. B., Transcendental Numbers, Nauka, Moscow, 1987 [Translated from the Russian by Neal Koblitz. de Gruyter Studies in Mathematics, 12, Walter de Gruyter & Co., Berlin-New York, 1989].Google Scholar