# of squares

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## 1—10 of 80 matching pages

##### 1: 27.17 Other Applications

##### 2: 27.13 Functions

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►The basic problem is that of expressing a given positive integer $n$ as a sum of integers from some prescribed set $S$ whose members are primes, squares, cubes, or other special integers.
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###### §27.13(iv) Representation by Squares

►For a given integer $k\ge 2$ the function ${r}_{k}\left(n\right)$ is defined as the number of solutions of the equation … ►Jacobi (1829) notes that ${r}_{2}\left(n\right)$ is the coefficient of ${x}^{n}$ in the square of the theta function $\vartheta \left(x\right)$: … ►For more than 8 squares, Milne’s identities are not the same as those obtained earlier by Mordell and others.##### 3: 8.23 Statistical Applications

###### §8.23 Statistical Applications

… ►Particular forms are the chi-square distribution functions; see Johnson et al. (1994, pp. 415–493). …##### 4: 19.38 Approximations

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►Approximations for Legendre’s complete or incomplete integrals of all three kinds, derived by Padé approximation of the square root in the integrand, are given in Luke (1968, 1970).
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##### 5: 27.22 Software

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##### 6: 34.14 Tables

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►Tables of exact values of the squares of the $\mathit{3}j$ and $\mathit{6}j$ symbols in which all parameters are $\le 8$ are given in Rotenberg et al. (1959), together with a bibliography of earlier tables of $\mathit{3}j,\mathit{6}j$, and $\mathit{9}j$ symbols on pp.
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##### 7: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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###### §1.18(ii) ${{L}}^{{2}}$ spaces on intervals in ${\mathbb{R}}$

►Let ${X}{=}{[}{a}{,}{b}{]}$ or ${[}{a}{,}{b}{)}$ or ${(}{a}{,}{b}{]}$ or ${(}{a}{,}{b}{)}$ be a (possibly infinite, or semi-infinite) interval in ${\mathbb{R}}$ . For a Lebesgue–Stieltjes measure ${d}{\alpha}$ on ${X}$ let ${{L}}^{{2}}{}{(}{X}{,}{d}{\alpha}{)}$ be the space of all Lebesgue–Stieltjes measurable complex-valued functions on ${X}$ which are square integrable with respect to ${d}{\alpha}$ , …Functions ${f}{,}{g}{\in}{{L}}^{{2}}{}{(}{X}{,}{d}{\alpha}{)}$ for which ${\u27e8}{f}{-}{g}{,}{f}{-}{g}{\u27e9}{=}{0}$ are identified with each other. The space ${{L}}^{{2}}{}{(}{X}{,}{d}{\alpha}{)}$ becomes a separable Hilbert space with inner product … ►Assume that ${{\left\{}{{\varphi}}_{{n}}{\right\}}}_{{n}{=}{0}}^{{\mathrm{\infty}}}$ is an orthonormal basis of ${{L}}^{{2}}{}{\left(}{X}{\right)}$ . The formulas in §1.18(i) are then: … ►for ${f}{}{(}{x}{)}{\in}{{L}}^{{2}}$ and piece-wise continuous, with convergence as discussed in §1.18(ii). …##### 8: 18.36 Miscellaneous Polynomials

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►These are matrix-valued polynomials that are orthogonal with respect to a square matrix of measures on the real line.
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##### 9: 3.11 Approximation Techniques

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###### §3.11(v) Least Squares Approximations

… ► … ► … ► … ►For further information on least squares approximations, including examples, see Gautschi (1997a, Chapter 2) and Björck (1996, Chapters 1 and 2). …##### 10: 19.31 Probability Distributions

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${R}_{G}(x,y,z)$ and ${R}_{F}(x,y,z)$ 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.
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