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zeros of classical orthogonal polynomials

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1: 18.16 Zeros
Inequalities
Asymptotic Behavior
when α ( - 1 2 , 1 2 ) . … Lastly, in view of (18.7.19) and (18.7.20), results for the zeros of L n ( ± 1 2 ) ( x ) lead immediately to results for the zeros of H n ( x ) . … For further information on the zeros of the classical orthogonal polynomials, see Szegő (1975, Chapter VI), Erdélyi et al. (1953b, §§10.16 and 10.17), Gatteschi (1987, 2002), López and Temme (1999a), and Temme (1990a).
2: 18.2 General Orthogonal Polynomials
§18.2(vi) Zeros
3: 18.39 Physical Applications
For physical applications of q -Laguerre polynomials see §17.17. …
4: Bibliography D
  • D. K. Dimitrov and G. P. Nikolov (2010) Sharp bounds for the extreme zeros of classical orthogonal polynomials. J. Approx. Theory 162 (10), pp. 1793–1804.
  • K. Driver and K. Jordaan (2013) Inequalities for extreme zeros of some classical orthogonal and q -orthogonal polynomials. Math. Model. Nat. Phenom. 8 (1), pp. 48–59.
  • 5: 18.38 Mathematical Applications
    §18.38(i) Classical OP’s: Numerical Analysis
    Quadrature
    Integrable Systems
    Riemann–Hilbert Problems
    Radon Transform
    6: Bibliography I
  • M. E. H. Ismail (2000a) An electrostatics model for zeros of general orthogonal polynomials. Pacific J. Math. 193 (2), pp. 355–369.
  • M. E. H. Ismail (2000b) More on electrostatic models for zeros of orthogonal polynomials. Numer. Funct. Anal. Optim. 21 (1-2), pp. 191–204.
  • M. E. H. Ismail (2005) Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge.
  • M. E. H. Ismail (2009) Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge.
  • M. E. H. Ismail and X. Li (1992) Bound on the extreme zeros of orthogonal polynomials. Proc. Amer. Math. Soc. 115 (1), pp. 131–140.
  • 7: 18.17 Integrals
    §18.17 Integrals
    §18.17(v) Fourier Transforms
    §18.17(vi) Laplace Transforms
    §18.17(vii) Mellin Transforms
    §18.17(ix) Compendia
    8: 18.41 Tables
    §18.41(i) Polynomials
    For P n ( x ) ( = P n ( x ) ) see §14.33. Abramowitz and Stegun (1964, Tables 22.4, 22.6, 22.11, and 22.13) tabulates T n ( x ) , U n ( x ) , L n ( x ) , and H n ( x ) for n = 0 ( 1 ) 12 . …
    §18.41(ii) Zeros
    For P n ( x ) , L n ( x ) , and H n ( x ) see §3.5(v). …
    9: 18.21 Hahn Class: Interrelations
    §18.21 Hahn Class: Interrelations
    §18.21(i) Dualities
    §18.21(ii) Limit Relations and Special Cases
    Hahn Jacobi
    Meixner Laguerre
    10: Bibliography B
  • E. Bannai (1990) Orthogonal Polynomials in Coding Theory and Algebraic Combinatorics. In Orthogonal Polynomials (Columbus, OH, 1989), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 294, pp. 25–53.
  • P. Baratella and L. Gatteschi (1988) The Bounds for the Error Term of an Asymptotic Approximation of Jacobi Polynomials. In Orthogonal Polynomials and Their Applications (Segovia, 1986), Lecture Notes in Math., Vol. 1329, pp. 203–221.
  • P. Bleher and A. Its (1999) Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model. Ann. of Math. (2) 150 (1), pp. 185–266.
  • R. Bo and R. Wong (1999) A uniform asymptotic formula for orthogonal polynomials associated with exp ( - x 4 ) . J. Approx. Theory 98, pp. 146–166.
  • C. Brezinski (1980) Padé-type Approximation and General Orthogonal Polynomials. International Series of Numerical Mathematics, Vol. 50, Birkhäuser Verlag, Basel.