monic polynomial

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1: 29.21 Tables

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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

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See accompanying text — Figure 18.4.7: Monic Hermite polynomials $h_{n}(x)=2^{-n}H_{n}\left(x\right)$ , $n=1,2,3,4,5$ . Magnify

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3: 3.5 Quadrature

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Gauss–Legendre Formula

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Gauss–Jacobi Formula

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Gauss–Laguerre Formula

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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

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§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: 18.30 Associated OP’s

… ►The lowest order monic versions of both of these appear in §18.2(x), (18.2.31) defining the

c=1

associated monic polynomials, and (18.2.32) their closely related cousins the

c=0

corecursive polynomials. … ►

§18.30(vii) Corecursive and Associated Monic Orthogonal Polynomials

… ►The simplicity of the relationship follows from the fact that the monic polynomials have been rescaled so that the coefficient of the highest power of

x

p_{n}(x)

, namely,

x^{n}

, is unity; for a note on this standardization, see §18.2(iii). … ►The zeroth order corecursive monic polynomials

\hat{p}_{n}^{(0)}(x)

follow directly from the alternate initialization … ►More generally, the

k

th corecursive monic polynomials (defined with the initialization of (18.30.28) followed by the

c=k

recurrence of (18.30.27)) are related to the

(k+1)

st monic associated polynomials by …

6: 18.38 Mathematical Applications

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Approximation Theory

►The monic Chebyshev polynomial

2^{1-n}T_{n}\left(x\right)

n\geq 1

, enjoys the ‘minimax’ property on the interval

[-1,1]

, that is,

|2^{1-n}T_{n}\left(x\right)|

has the least maximum value among all monic polynomials of degree

n

. …

7: 18.33 Polynomials Orthogonal on the Unit Circle

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§18.33(vi) Alternative Set-up with Monic Polynomials

►Instead of orthonormal polynomials

\{\phi_{n}(z)\}

Simon (2005a, b) uses monic polynomials

\Phi_{n}(z)

. …A system of monic polynomials

\{\Phi_{n}(z)\}

n=0,1,\dots

, where

\Phi_{n}(x)

is of proper degree

n

, is orthogonal on the unit circle with respect to the measure

\mu

if …

8: 18.2 General Orthogonal Polynomials

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§18.2(x) Orthogonal Polynomials and Continued Fractions

… ►Define the first associated monic orthogonal polynomials

p_{n}^{(1)}(x)

as monic OP’s satisfying … ►The

p_{n}^{(0)}(x)

are the monic corecursive orthogonal polynomials. … ►Because of (18.2.36) the OP’s

p_{n}(x)

are also called monic denominator polynomials and the OP’s

p_{n-1}^{(1)}(x)

, or, equivalently, the

p_{n}^{(0)}(x)

, are called the monic numerator polynomials. …

9: 18.35 Pollaczek Polynomials

… ►For the monic polynomials … ►More generally, the

P^{(\lambda)}_{n}\left(x;a,b\right)

are OP’s if and only if one of the following three conditions holds (in case (iii) work with the monic polynomials (18.35.2_2)). …

10: 32.8 Rational Solutions

… ►where the

Q_{n}(z)

are monic polynomials (coefficient of highest power of

z

1

) satisfying …