# 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 ${w(x)=\exp\left(-Q(x)\right)},$ $-\infty,
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). …
18.32.2 $w(x)={\left|x\right|}^{\alpha}\exp\left(-Q(x)\right),$ $x\in\mathbb{R}$, $\alpha>-1$,
##### 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 $\{p_{n}(x)\}$, $n=0,1,\dots$, 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
For $-1-\beta>\alpha>-1$ a finite system of Jacobi polynomials $P^{(\alpha,\beta)}_{n}\left(x\right)$ is orthogonal on $(1,\infty)$ with weight function $w(x)=(x-1)^{\alpha}(x+1)^{\beta}$. …
##### 6: 18.39 Applications in the Physical Sciences
###### §18.39(iii) Non Classical WeightFunctions of Utility in DVR Method in the Physical Sciences
Shizgal (2015, Chapter 2), contains a broad-ranged discussion of methods and applications for these, and other, non-classical weight functions. … Graphs of the weight functions of (18.39.50) are shown in Figure 18.39.2. … In the attractive case (18.35.6_4) for the discrete parts of the weight function where with $x_{k}<-1$, are also simplified: …
##### 7: 18.33 Polynomials Orthogonal on the Unit Circle
A system of polynomials $\{\phi_{n}(z)\}$, $n=0,1,\dots$, where $\phi_{n}(z)$ is of proper degree $n$, is orthonormal on the unit circle with respect to the weight function $w(z)$ ($\geq 0$) if … 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 $\phi_{2n}(z^{\pm 1})$ or $\phi_{2n+1}(z^{\pm 1})$. …
18.33.19 $\,\mathrm{d}\mu(z)=\frac{1}{2\pi\mathrm{i}}\,w(z)\frac{\,\mathrm{d}z}{z}$
for some weight function $w(z)$ ($\geq 0$) then (18.33.17) (see also (18.33.1)) takes the form … For $w(z)$ as in (18.33.19) (or more generally as the weight function of the absolutely continuous part of the measure $\mu$ in (18.33.17)) and with $\alpha_{n}$ the Verblunsky coefficients in (18.33.23), (18.33.24), Szegő’s theorem states that …
##### 8: 18.2 General Orthogonal Polynomials
A system (or set) of polynomials $\{p_{n}(x)\}$, $n=0,1,2,\ldots$, where $p_{n}(x)$ has degree $n$ as in §18.1(i), is said to be orthogonal on $(a,b)$ with respect to the weight function $w(x)$ ($\geq 0$) ifFor OP’s $\{p_{n}(x)\}$ on $\mathbb{R}$ with respect to an even weight function $w(x)$ we have … Under further conditions on the weight function there is an equiconvergence theorem, see Szegő (1975, Theorem 13.1.2). …
##### 9: 18.40 Methods of Computation
18.40.4 $\lim_{N\to\infty}F_{N}(z)=F(z)\equiv\frac{1}{\mu_{0}}\int_{a}^{b}\frac{w(x)\,% \mathrm{d}x}{z-x},$ $z\in\mathbb{C}\backslash[a,b]$,
18.40.6 $\lim_{\varepsilon\to 0{+}}\int_{a}^{b}\frac{w(x)\,\mathrm{d}x}{x^{\prime}+% \mathrm{i}\varepsilon-x}\,\mathrm{d}x=\pvint_{a}^{b}\frac{w(x)\,\mathrm{d}x}{x% ^{\prime}-x}-\mathrm{i}\pi w(x^{\prime}),$
18.40.8 $w(x_{i,N})\approx\frac{w_{i,N}}{\left.\frac{\mathrm{d}x(j,N)}{\mathrm{d}j}% \right|_{j=i}}.$
The example chosen is inversion from the $\alpha_{n},\beta_{n}$ for the weight function for the repulsive Coulomb–Pollaczek, RCP, polynomials of (18.39.50). …
##### 10: 18.19 Hahn Class: Definitions
18.19.2 $w(z;a,b,\overline{a},\overline{b})=\Gamma\left(a+iz\right)\Gamma\left(b+iz% \right)\Gamma\left(\overline{a}-iz\right)\Gamma\left(\overline{b}-iz\right),$
18.19.3 $w(x)=w(x;a,b,\overline{a},\overline{b})=|\Gamma\left(a+\mathrm{i}x\right)% \Gamma\left(b+\mathrm{i}x\right)|^{2},$
18.19.7 $w^{(\lambda)}(z;\phi)=\Gamma\left(\lambda+iz\right)\Gamma\left(\lambda-iz% \right){\mathrm{e}}^{(2\phi-\pi)z},$