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1: 18.32 OP’s with Respect to Freud Weights

§18.32 OP’s with Respect to Freud Weights

►A Freud weight is a weight function of the form ►

18.32.1 $$w(x)=\exp \left(-Q(x)\right),$$ ,

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Symbols:

$\exp z$: exponential function, $w(x)$: weight function and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.32.E1

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Annotations for §18.32 and Ch.18

… ►For asymptotic approximations to OP’s that correspond to Freud weights with more general functions $Q(x)$ see Deift et al. (1999a, b), Bleher and Its (1999), and Kriecherbauer and McLaughlin (1999).

2: 3.5 Quadrature

… ►

Gauss Formula for a Logarithmic Weight Function

… ►

Table 3.5.14: Nodes and weights for the 5-point Gauss formula for the logarithmic weight function.

► ►►

$x_k$		$w_k$
…

►

►

Table 3.5.15: Nodes and weights for the 10-point Gauss formula for the logarithmic weight function.

► ►►

$x_k$		$w_k$
…

►

… ►

Table 3.5.17: Nodes and weights for the 20-point Gauss formula for the logarithmic weight function.

► ►►

$x_k$		$w_k$
…

►

… ►Then the weights are given by …

3: 1.2 Elementary Algebra

… ►

§1.2(iv) Means

… ►If $r$ is a nonzero real number, then the weighted mean $M(r)$ of $n$ nonnegative numbers $a_1,a_2,\mathrm{\dots },a_n$, and $n$ positive numbers $p_1,p_2,\mathrm{\dots },p_n$ with … ►

1.2.22 $$M(r)=0,$$ and $a_1{a}_2\mathrm{\dots }{a}_n=0$.

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Symbols:

$n$: nonnegative integer and $M(r)$: weighted mean

A&S Ref:

3.1.15

Permalink:

http://dlmf.nist.gov/1.2.E22

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Annotations for §1.2(iv), §1.2 and Ch.1

… ►

$M(1)=A,$

►

$M(-1)=H,$

…

4: 18.36 Miscellaneous Polynomials

… ►These are OP’s on the interval $(-1,1)$ with respect to an orthogonality measure obtained by adding constant multiples of “Dirac delta weights” at $-1$ and $1$ to the weight function for the Jacobi polynomials. …

5: 12.15 Generalized Parabolic Cylinder Functions

… ►This equation arises in the study of non-self-adjoint elliptic boundary-value problems involving an indefinite weight function. …

6: 18.2 General Orthogonal Polynomials

… ►A system (or set) of polynomials $\{p_n(x)\}$, $n=0,1,2,\mathrm{\dots }$, is said to be orthogonal on $(a,b)$ with respect to the weight function $w(x)$ ($\ge 0$) if … ►Then a system of polynomials $\{p_n(x)\}$, $n=0,1,2,\mathrm{\dots }$, is said to be orthogonal on $X$ with respect to the weights $w_x$ if … ►

18.2.4 $n=0,1,\mathrm{\dots }$,

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Symbols:

$\in$: element of, $w_x$: weights, $X$: subset of $ℝ$ , $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.2.E4

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Annotations for §18.2(i), §18.2(i), §18.2 and Ch.18

… ►

18.2.5 $$h_n={\int }_a^b{\left(p_n(x)\right)}^{2}w(x)dx\text{ or }\sum _{x\in X}{\left(p_n(x)\right)}^2{w}_x,$$

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Defines:

$h_n$ (locally)

Symbols:

$dx$: differential, $\in$: element of, $\int$: integral, $w(x)$: weight function, $w_x$: weights, $X$: subset of $ℝ$ , $n$: nonnegative integer, $p_n(x)$: polynomial of degree $n$ and $x$: real variable

A&S Ref:

22.1.2

Referenced by:

§3.5(v)

Permalink:

http://dlmf.nist.gov/18.2.E5

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Annotations for §18.2(iii), §18.2 and Ch.18

►

18.2.6 $${\tilde{h}}_n={\int }_a^{b}x{\left(p_n(x)\right)}^{2}w(x)dx\text{ or }\sum _{x\in X}x{\left(p_n(x)\right)}^2{w}_x,$$

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Defines:

${\tilde{h}}_n$ (locally)

Symbols:

$dx$: differential, $\in$: element of, $\int$: integral, $w(x)$: weight function, $w_x$: weights, $X$: subset of $ℝ$ , $n$: nonnegative integer, $p_n(x)$: polynomial of degree $n$ and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.2.E6

Encodings:

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Annotations for §18.2(iii), §18.2 and Ch.18

…

7: 18.31 Bernstein–Szegő Polynomials

… ►The Bernstein–Szegő polynomials $\{p_n(x)\}$, $n=0,1,\mathrm{\dots }$, are orthogonal on $(-1,1)$ with respect to three types of weight function: ${(1-x^2)}^{-\frac{1}{2}}{(\rho (x))}^{-1}$, ${(1-x^2)}^{\frac{1}{2}}{(\rho (x))}^{-1}$, ${(1-x)}^{\frac{1}{2}}{(1+x)}^{-\frac{1}{2}}{(\rho (x))}^{-1}$. …

8: 1.7 Inequalities

… ►

1.7.8 $$\min (a_1,a_2,\mathrm{\dots },a_n)\le M(r)\le \max (a_1,a_2,\mathrm{\dots },a_n),$$

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Symbols:

$n$: nonnegative integer and $M(r)$: weighted mean

A&S Ref:

3.2.2

Permalink:

http://dlmf.nist.gov/1.7.E8

Encodings:

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Annotations for §1.7(iii), §1.7 and Ch.1

… ►

1.7.9 $$M(r)\le M(s),$$ ,

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Symbols:

$M(r)$: weighted mean

A&S Ref:

3.2.4

Permalink:

http://dlmf.nist.gov/1.7.E9

Encodings:

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Annotations for §1.7(iii), §1.7 and Ch.1

…

9: 18.3 Definitions

… ►

Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, normalizations, leading coefficients, and parameter constraints. …

► ►►

Name	$p_n(x)$	$(a,b)$	$w(x)$	$h_n$	$k_n$	${\tilde{k}}_n/k_n$	Constraints
…

►

…

10: 18.25 Wilson Class: Definitions

… ►If $\alpha +1=-N$, then the weights will be positive iff one of the following eight sets of inequalities holds: … ►

§18.25(ii) Weights and Normalizations: Continuous Cases

►

18.25.2 $${\int }_0^{\mathrm{\infty }}{p}_n(x)p_m(x)w(x)dx=h_n{\delta }_{n,m}.$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $dx$: differential, $\int$: integral, $w(x)$: weight function, $m$: nonnegative integer, $n$: nonnegative integer, $p_n(x)$: polynomial of degree $n$ , $x$: real variable and $h_n$

Permalink:

http://dlmf.nist.gov/18.25.E2

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Annotations for §18.25(ii), §18.25 and Ch.18

… ►

18.25.4 $$w(y^2)=\frac{1}{2y}{\left|\frac{{\prod }_j\mathrm{\Gamma }\left(a_j+\mathrm{i}y\right)}{\mathrm{\Gamma }\left(2\mathrm{i}y\right)}\right|}^2,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{i}$: imaginary unit, $y$: real variable and $w(x)$: weight function

Permalink:

http://dlmf.nist.gov/18.25.E4

Encodings:

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See also:

Annotations for §18.25(ii), §18.25(ii), §18.25 and Ch.18

… ►

§18.25(iii) Weights and Normalizations: Discrete Cases

…