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Symbolic computation of recurrence coefficients for polynomials orthogonal with respect to the Szegő-Bernstein weights

  • Corresponding author: Gradimir V. Milovanović

    Corresponding author: Gradimir V. Milovanović

Dedicated to occasion of 60th birthday of Prof. Vijay Gupta

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  • The coefficients in the three-term recurrence relation for monic orthogonal polynomials with respect to the Szegő-Bernstein weight functions wν(x)=W(x)/(cx)ν, ν1, on (1,1) are obtained in the explicit form for all Chebyshev cases, i.e., when W(x) is (1x2)1/2, (1+x)/(1x) and (1x)/(1+x). Chebyshev's method of modified moments is used, as well as the MATHEMATICA package OrthogonalPolinomials developed in [Facta Univ. Ser. Math. Inform. 19 (2004), 17-36] and [Math. Balkanica 26 (2012), 169-184].

    Mathematics Subject Classification: Primary: 33C45; Secondary: 33F10.

    Citation:

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  • Figure 1.  Three-term recurrence coefficients αk (left) and βk (right) for ν=4

    Figure 2.  Three-term recurrence coefficients αk (left) and βk (right) for ν=5

    Figure 3.  Three-term recurrence coefficients αk (left) and βk (right) for ν=10

    Figure 4.  Three-term recurrence coefficients αk (left) and βk (right) for ν=5

    Figure 5.  Three-term recurrence coefficients αk (left) and βk (right) for ν=10

    Figure 6.  Three-term recurrence coefficients αk (left) and βk (right) for ν=10

    Figure 7.  Three-term recurrence coefficients αk (left) and βk (right) for ν=10

  • [1] G. E. Andrews, R. Askey and R. Roy, Special Functions, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9781107325937.
    [2] P. L. Chebyshev, Sur l'interpolation par la méthode des moindres carrés, Mém. Acad. Impér. Sci. St. Petersbourg (7), 1 (1859), 1-24. 
    [3] A. S. Cvetković and G. V. Milovanović, The mathematica package "OrthogonalPolynomials", Facta Univ. Ser. Math. Inform., 19 (2004), 17-36. 
    [4] B. Fischer and G. Golub, How to generate unknown orthogonal polynomials out of known orthogonal polynomials, J. Comput. Appl. Math., 43 (1992), 99-115.  doi: 10.1016/0377-0427(92)90261-U.
    [5] W. Gautschi, On generating orthogonal polynomials, SIAM J. Sci. Statist. Comput., 3 (1982), 289-317.  doi: 10.1137/0903018.
    [6] W. Gautschi, Orthogonal polynomials: Applications and computation, Acta Numerica, 5 (1996), 45-119.  doi: 10.1017/S0962492900002622.
    [7] W. Gautschi, Orthogonal Polynomials: Computation and Approximation, Oxford Science Publications. Oxford University Press, New York, 2004.
    [8] W. Gautschi and G. V. Milovanović, Orthogonal polynomials relative to a generalized Marchenko-Pastur probability measure, Numer. Algorithms, 88 (2021), 1233-1249.  doi: 10.1007/s11075-021-01073-1.
    [9] W. Gautschi and S. E. Notaris, Gauss–Kronrod quadrature formulae for weight functions of Bernstein–Szegö type, J. Comput. Appl. Math., 25 (1989), 199-224.  doi: 10.1016/0377-0427(89)90047-2.
    [10] G. Mastroianni and G. V. Milovanović, Weighted integration of periodic functions on the real line, Appl. Math. Comput., 128 (2002), 365-378.  doi: 10.1016/S0096-3003(01)00080-7.
    [11] G. Mastroianni and G. V. Milovanović, Interpolation Processes – Basic Theory and Applications, Springer Monographs in Mathematics, Springer – Verlag, Berlin – Heidelberg, 2008. doi: 10.1007/978-3-540-68349-0.
    [12] G. V. Milovanović and A. S. Cvetković, Special classes of orthogonal polynomials and corresponding quadratures of Gaussian type, Math. Balkanica, 26 (2012), 169-184. 
    [13] S. E. Notaris, Gauss–Kronrod quadrature formulae for weight functions of Bernstein–Szegö type, II, J. Comput. Appl. Math., 29 (1990), 161-169.  doi: 10.1016/0377-0427(90)90355-4.
    [14] H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier, Inc., Amsterdam, 2012. doi: 10.1016/B978-0-12-385218-2.00001-3.
    [15] G. Szegő, Orthogonal Polynomials, American Mathematical Society Colloquium Publications, Vol. 23 American Mathematical Society, Providence, R.I. 1959
    [16] R. Wong and Y.-Q. Zhao, Recent advances in asymptotic analysis, Anal. Appl. (Singap.), 2022. doi: 10.1142/S0219530522400012.
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