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1: 29.21 Tables
  • Arscott and Khabaza (1962) tabulates the coefficients of the polynomials P in Table 29.12.1 (normalized so that the numerically largest coefficient is unity, i.e. monic polynomials), and the corresponding eigenvalues h for k 2 = 0.1 ( .1 ) 0.9 , n = 1 ( 1 ) 30 . Equations from §29.6 can be used to transform to the normalization adopted in this chapter. Precision is 6S.

  • 2: 18.4 Graphics
    See accompanying text
    Figure 18.4.7: Monic Hermite polynomials h n ( x ) = 2 - n H n ( x ) , n = 1 , 2 , 3 , 4 , 5 . Magnify
    3: 3.5 Quadrature
    Gauss–Legendre Formula
    Gauss–Jacobi Formula
    Gauss–Laguerre Formula
    Gauss–Hermite Formula
    All the monic orthogonal polynomials { p n } used with Gauss quadrature satisfy a three-term recurrence relation (§18.2(iv)): …
    4: 1.11 Zeros of Polynomials
    §1.11(ii) Elementary Properties
    Every monic (coefficient of highest power is one) polynomial of odd degree with real coefficients has at least one real zero with sign opposite to that of the constant term. A monic polynomial of even degree with real coefficients has at least two zeros of opposite signs when the constant term is negative. …
    5: 18.38 Mathematical Applications
    The scaled Chebyshev polynomial 2 1 - n T n ( x ) , n 1 , enjoys the “minimax” property on the interval [ - 1 , 1 ] , that is, | 2 1 - n T n ( x ) | has the least maximum value among all monic polynomials of degree n . …
    6: 18.2 General Orthogonal Polynomials
    then two special normalizations are: (i) orthonormal OP’s: h n = 1 , k n > 0 ; (ii) monic OP’s: k n = 1 . …
    7: 32.8 Rational Solutions
    where the Q n ( z ) are monic polynomials (coefficient of highest power of z is 1 ) satisfying …