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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

…

3: 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

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Associated Monic OP’s

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The “Zeroth” Corecursive Monic OP

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Relationship of Monic Corecursive and Monic Associated OP’s

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4: 18.2 General Orthogonal Polynomials

… ►(ii) monic OP’s:

k_{n}=1

. … ►

Monic and Orthonormal Forms

… ►the monic recurrence relations (18.2.8) and (18.2.10) take the form … ►Define the first associated monic orthogonal polynomials

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

as monic OP’s satisfying …

5: 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)): …

6: 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. …

7: 18.33 Polynomials Orthogonal on the Unit Circle

… ►Simon (2005a, b) gives the general theory of these OP’s in terms of monic OP’s

\Phi_{n}(x)

, see §18.33(vi). … ►Instead of (18.33.9) one might take monic OP’s

\{q_{n}(x)\}

with weight function

(1+x)w_{1}(x)

, and then express

q_{n}\left(\tfrac{1}{2}(z+z^{-1})\right)

in terms of

\phi_{2n}(z^{\pm 1})

\phi_{2n+1}(z^{\pm 1})

. … ►

§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: 18.38 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

. …

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 …