weight functions

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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: 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. …

3: 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: ${(1-x^2)}^{-\frac{1}{2}}{(\rho (x))}^{-1}$, ${(1-x^2)}^{\frac{1}{2}}{(\rho (x))}^{-1}$, ${(1-x)}^{\frac{1}{2}}{(1+x)}^{-\frac{1}{2}}{(\rho (x))}^{-1}$. …

4: 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
…

►

…

5: 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 ►

18.2.1 $${\int }_a^b{p}_n(x)p_m(x)w(x)dx=0,$$ $n\ne m$.

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Symbols:

$dx$: differential, $\int$: integral, $w(x)$: weight function, $m$: nonnegative integer, $n$: nonnegative integer, $p_n(x)$: polynomial of degree $n$ , $a$: interval endpoint, $b$: interval endpoint and $x$: real variable

A&S Ref:

22.1.1

Referenced by:

§18.2(i), §18.2(iii), §18.5(iii)

Permalink:

http://dlmf.nist.gov/18.2.E1

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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

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

…

6: 3.5 Quadrature

… ►An interpolatory quadrature rule … ►

Gauss Formula for a Logarithmic Weight Function

… ►

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

► ►►

$x_k$		$w_k$
…

►

… ►

Table 3.5.18: Nodes and weights for the 5-point complex Gauss quadrature formula with $s=1$.

► ►►

${\zeta }_k$			$w_k$
…

►

… ►

Table 3.5.21: Cubature formulas for disk and square.

► ►►

Diagram	$(x_j,y_j)$	$w_j$	$R$
…

►

7: 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. …

8: 18.19 Hahn Class: Definitions

… ►

Table 18.19.1: Orthogonality properties for Hahn, Krawtchouk, Meixner, and Charlier OP’s: discrete sets, weight functions, normalizations, and parameter constraints.

► ►►

$p_n(x)$	$X$	$w_x$	$h_n$
…

►

… ►

18.19.2 $$w(z;a,b,\overline{a},\overline{b})=\mathrm{\Gamma }\left(a+\mathrm{i}z\right)\mathrm{\Gamma }\left(b+\mathrm{i}z\right)\mathrm{\Gamma }\left(\overline{a}-\mathrm{i}z\right)\mathrm{\Gamma }\left(\overline{b}-\mathrm{i}z\right),$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/18.19.E2

Encodings:

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See also:

Annotations for §18.19, §18.19 and Ch.18

►

18.19.3 $$w(x)=w(x;a,b,\overline{a},\overline{b})={|\mathrm{\Gamma }\left(a+\mathrm{i}x\right)\mathrm{\Gamma }\left(b+\mathrm{i}x\right)|}^2,$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/18.19.E3

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Annotations for §18.19, §18.19 and Ch.18

… ►

18.19.7 $$w^{(\lambda )}(z;\varphi )=\mathrm{\Gamma }\left(\lambda +\mathrm{i}z\right)\mathrm{\Gamma }\left(\lambda -\mathrm{i}z\right){\mathrm{e}}^{(2\varphi -\pi )z},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $w(x)$: weight function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/18.19.E7

Encodings:

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See also:

Annotations for §18.19, §18.19 and Ch.18

…

9: 18.33 Polynomials Orthogonal on the Unit Circle

… ►A system of polynomials $\{{\varphi }_n(z)\}$, $n=0,1,\mathrm{\dots }$, where ${\varphi }_n(z)$ is of proper degree $n$, is orthonormal on the unit circle with respect to the weight function $w(z)$ ($\ge 0$) if ►

18.33.1 $$\frac{1}{2\pi \mathrm{i}}{\int }_{|z|=1}{\varphi }_n(z)\overline{{\varphi }_m(z)}w(z)\frac{dz}{z}={\delta }_{n,m},$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $dx$: differential, $\mathrm{i}$: imaginary unit, $\int$: integral, $w(x)$: weight function, $z$: complex variable, $m$: nonnegative integer, $n$: nonnegative integer and ${\varphi }_n(z)$: polynomials

Permalink:

http://dlmf.nist.gov/18.33.E1

Encodings:

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See also:

Annotations for §18.33(i), §18.33 and Ch.18

… ►Let $\{p_n(x)\}$ and $\{q_n(x)\}$, $n=0,1,\mathrm{\dots }$, be OP’s with weight functions $w_1(x)$ and $w_2(x)$, respectively, on $(-1,1)$. … ►

18.33.16 $$w(z)={\left|{(qz;q^2)}_{\mathrm{\infty }}/{(aqz;q^2)}_{\mathrm{\infty }}\right|}^2,$$ .

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Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $w(x)$: weight function, $z$: complex variable and $q$: real variable

Referenced by:

§18.33(v)

Permalink:

http://dlmf.nist.gov/18.33.E16

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See also:

Annotations for §18.33(iv), §18.33(iv), §18.33 and Ch.18

…

10: 18.25 Wilson Class: Definitions

… ►

§18.25(ii) Weights and Normalizations: Continuous Cases

►

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

ⓘ

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

Encodings:

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See also:

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

… ►

18.25.7 $$w(y^2)=\frac{1}{2y}{\left|\frac{{\prod }_j\mathrm{\Gamma }\left(a_j+\mathrm{i}y\right)}{\mathrm{\Gamma }\left(2\mathrm{i}y\right)}\right|}^2,$$

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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.E7

Encodings:

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See also:

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

… ►

18.25.15 $$h_n=\frac{n!(N-n)!{\left(\gamma +\delta +2\right)}_N}{N!{\left(\gamma +1\right)}_n{\left(\delta +1\right)}_{N-n}}.$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $n$: nonnegative integer, $N$: positive integer, $\delta$: arbitary small positive constant and $h_n$

Referenced by:

§18.25(iii)

Permalink:

http://dlmf.nist.gov/18.25.E15

Encodings:

pMML, png, TeX

See also:

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

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