# 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).
##### 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. …
##### 4: 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: $(1-x^{2})^{-\frac{1}{2}}(\rho(x))^{-1}$, $(1-x^{2})^{\frac{1}{2}}(\rho(x))^{-1}$, $(1-x)^{\frac{1}{2}}(1+x)^{-\frac{1}{2}}(\rho(x))^{-1}$. …
##### 6: 18.2 General Orthogonal Polynomials
A system (or set) of polynomials $\{p_{n}(x)\}$, $n=0,1,2,\ldots$, is said to be orthogonal on $(a,b)$ with respect to the weight function $w(x)$ ($\geq 0$) if
18.2.1 $\int_{a}^{b}p_{n}(x)p_{m}(x)w(x)\mathrm{d}x=0,$ $n\neq m$.
##### 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
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 $w^{(\lambda)}(z;\phi)=\Gamma\left(\lambda+iz\right)\Gamma\left(\lambda-iz% \right)e^{(2\phi-\pi)z},$
##### 9: 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
18.33.1 $\frac{1}{2\pi\mathrm{i}}\int_{|z|=1}\phi_{n}(z)\overline{\phi_{m}(z)}w(z)\frac% {\mathrm{d}z}{z}=\delta_{n,m},$
Let $\{p_{n}(x)\}$ and $\{q_{n}(x)\}$, $n=0,1,\dots$, be OP’s with weight functions $w_{1}(x)$ and $w_{2}(x)$, respectively, on $(-1,1)$. …
##### 10: 18.25 Wilson Class: Definitions
###### §18.25(ii) Weights and Normalizations: Continuous Cases
18.25.2 $\int_{0}^{\infty}p_{n}(x)p_{m}(x)w(x)\mathrm{d}x=h_{n}\delta_{n,m}.$
18.25.4 $w(y^{2})=\frac{1}{2y}\left|\frac{\prod_{j}\Gamma\left(a_{j}+iy\right)}{\Gamma% \left(2iy\right)}\right|^{2},$
18.25.7 $w(y^{2})=\frac{1}{2y}\left|\frac{\prod_{j}\Gamma\left(a_{j}+iy\right)}{\Gamma% \left(2iy\right)}\right|^{2},$
18.25.15 $h_{n}=\frac{n!\,(N-n)!\,{\left(\gamma+\delta+2\right)_{N}}}{N!\,{\left(\gamma+% 1\right)_{n}}{\left(\delta+1\right)_{N-n}}}.$