# normal forms

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##### 11: Bibliography L
• G. Labahn and M. Mutrie (1997) Reduction of Elliptic Integrals to Legendre Normal Form. Technical report Technical Report 97-21, Department of Computer Science, University of Waterloo, Waterloo, Ontario.
• ##### 12: 3.6 Linear Difference Equations
The normalizing factor $\Lambda$ can be the true value of $w_{0}$ divided by its trial value, or $\Lambda$ can be chosen to satisfy a known property of the wanted solution of the formLet us assume the normalizing condition is of the form $w_{0}=\lambda$, where $\lambda$ is a constant, and then solve the following tridiagonal system of algebraic equations for the unknowns $w_{1}^{(N)},w_{2}^{(N)},\dots,w_{N-1}^{(N)}$; see §3.2(ii). … For further information, including a more general form of normalizing condition, other examples, convergence proofs, and error analyses, see Olver (1967a), Olver and Sookne (1972), and Wimp (1984, Chapter 6). …
##### 13: 31.15 Stieltjes Polynomials
The normalized system of products (31.15.8) forms an orthonormal basis in the Hilbert space $L_{\rho}^{2}(Q)$. …
##### 14: 33.2 Definitions and Basic Properties
The normalizing constant $C_{\ell}\left(\eta\right)$ is always positive, and has the alternative form
##### 15: 8.23 Statistical Applications
Particular forms are the chi-square distribution functions; see Johnson et al. (1994, pp. 415–493). The function $\mathrm{B}_{x}\left(a,b\right)$ and its normalization $I_{x}\left(a,b\right)$ play a similar role in statistics in connection with the beta distribution; see Johnson et al. (1995, pp. 210–275). …
##### 16: 30.15 Signal Analysis
The sequence $\phi_{n}$, $n=0,1,2,\dots$ forms an orthonormal basis in the space of $\sigma$-bandlimited functions, and, after normalization, an orthonormal basis in $L^{2}(-\tau,\tau)$. …
##### 17: 33.5 Limiting Forms for Small $\rho$, Small $|\eta|$, or Large $\ell$
###### §33.5(i) Small $\rho$
33.5.6 $C_{\ell}\left(0\right)=\frac{2^{\ell}\ell!}{(2\ell+1)!}=\frac{1}{(2\ell+1)!!}.$
##### 19: 19.31 Probability Distributions
$R_{G}\left(x,y,z\right)$ and $R_{F}\left(x,y,z\right)$ occur as the expectation values, relative to a normal probability distribution in ${\mathbb{R}}^{2}$ or ${\mathbb{R}}^{3}$, of the square root or reciprocal square root of a quadratic form. …
##### 20: 28.12 Definitions and Basic Properties
As in §28.7 values of $q$ for which (28.2.16) has simple roots $\lambda$ are called normal values with respect to $\nu$. For real values of $\nu$ and $q$ all the $\lambda_{\nu}\left(q\right)$ are real, and $q$ is normal. … If $q$ is a normal value of the corresponding equation (28.2.16), then these functions are uniquely determined as analytic functions of $z$ and $q$ by the normalization