sums of squares

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11: 27.13 Functions

… ►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. … ►This problem is named after Edward Waring who, in 1770, stated without proof and with limited numerical evidence, that every positive integer $n$ is the sum of four squares, of nine cubes, of nineteen fourth powers, and so on. … ►

27.13.5 $${(\vartheta \left(x\right))}^2=1+\sum _{n=1}^{\mathrm{\infty }}{r}_2\left(n\right)x^n.$$

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Symbols:

$\vartheta \left(x\right)={\theta }_3(0,x)$: alternative notation, $r_k\left(n\right)$: number of squares, $n$: positive integer and $x$: real number

Referenced by:

§27.13(iv)

Permalink:

http://dlmf.nist.gov/27.13.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §27.13(iv), §27.13 and Ch.27

… ►By similar methods Jacobi proved that $r_4\left(n\right)=8{\sigma }_1\left(n\right)$ if $n$ is odd, whereas, if $n$ is even, $r_4\left(n\right)=24$ times the sum of the odd divisors of $n$. …

12: 26.12 Plane Partitions

… ►where ${\sigma }_2(j)$ is the sum of the squares of the divisors of $j$. …

13: 18.39 Applications in the Physical Sciences

… ►The fact that non-$L^2$ continuum scattering eigenstates may be expressed in terms or (infinite) sums of $L^2$ functions allows a reformulation of scattering theory in atomic physics wherein no non-$L^2$ functions need appear. …

14: 18.18 Sums

… ►In all three cases of Jacobi, Laguerre and Hermite, if $f(x)$ is $L^2$ on the corresponding interval with respect to the corresponding weight function and if $a_n,b_n,d_n$ are given by (18.18.1), (18.18.5), (18.18.7), respectively, then the respective series expansions (18.18.2), (18.18.4), (18.18.6) are valid with the sums converging in $L^2$ sense. …

15: 27.6 Divisor Sums

… ►

27.6.1

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Symbols:

$\lambda \left(n\right)$: Liouville’s function, $d$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §27.6 and Ch.27

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16: 1.8 Fourier Series

… ►

1.8.5 $$\frac{1}{\pi }{\int }_{-\pi }^{\pi }{\left|f(x)\right|}^{2}dx=\frac{1}{2}{\left|a_0\right|}^2+\sum _{n=1}^{\mathrm{\infty }}({\left|a_n\right|}^2+{\left|b_n\right|}^2),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Referenced by:

Erratum (V1.2.0) for Equations (1.8.5), (1.8.6)

Permalink:

http://dlmf.nist.gov/1.8.E5

Encodings:

pMML, png, TeX

Modification (effective with 1.2.0):

Previously this equation was given as an inequality. For square integrable functions this inequality $\ge$ can be sharpened to $=$.

See also:

Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

►

1.8.6 $$\frac{1}{2\pi }{\int }_{-\pi }^{\pi }{\left|f(x)\right|}^{2}dx=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\left|c_n\right|}^2,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Referenced by:

§1.8(i), Erratum (V1.2.0) for Equations (1.8.5), (1.8.6)

Permalink:

http://dlmf.nist.gov/1.8.E6

Encodings:

pMML, png, TeX

Modification (effective with 1.2.0):

Previously this equation was given as an inequality. For square integrable functions this inequality $\ge$ can be sharpened to $=$.

See also:

Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

…

17: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►

1.18.14 $${\int }_a^b{\left|f(x)\right|}^{2}dx=\sum _{n=0}^{\mathrm{\infty }}{\left|c_n\right|}^2,$$ $f\in L^2\left(X\right)$,

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Symbols:

$L^2(X,dx)$: Lebesgue–Stieltjes measurable, square integrable, complex-valued functions, $dx$: differential of $x$ , $\in$: element of, $\int$: integral, $n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.18.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §1.18(ii), §1.18 and Ch.1

… ►

1.18.64 $$f(x)={\int }_{{𝝈}_c}\widehat{f}(\lambda ){\varphi }_{\lambda }(x)d\lambda +\sum _{{𝝈}_p}\widehat{f}({\lambda }_n){\varphi }_{{\lambda }_n}(x),$$ $f(x)\in C(X)\cap L^2\left(X\right)$.

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Symbols:

$L^2(X,dx)$: Lebesgue–Stieltjes measurable, square integrable, complex-valued functions, $dx$: differential of $x$ , $\in$: element of, $\int$: integral, $\cap$: intersection, $C(I)$ or $C(a,b)$: continuous on an interval $I$ or $(a,b)$ and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/1.18.E64

Encodings:

pMML, png, TeX

See also:

Annotations for §1.18(viii), §1.18 and Ch.1

… ►

1.18.65 $${\int }_X{\left|f(x)\right|}^{2}dx={\int }_{{𝝈}_c}{\left|\widehat{f}(\lambda )\right|}^{2}d\lambda +\sum _{{𝝈}_p}{\left|\widehat{f}({\lambda }_n)\right|}^2,$$ $f\in L^2\left(X\right)$.

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Symbols:

$L^2(X,dx)$: Lebesgue–Stieltjes measurable, square integrable, complex-valued functions, $dx$: differential of $x$ , $\in$: element of, $\int$: integral, $n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Referenced by:

§1.18(viii)

Permalink:

http://dlmf.nist.gov/1.18.E65

Encodings:

pMML, png, TeX

See also:

Annotations for §1.18(viii), §1.18 and Ch.1

… ►

1.18.66 $$⟨{\left(z-T\right)}^{-1}f,f⟩=\sum _{{𝝈}_p}\frac{{\left|\widehat{f}({\lambda }_n)\right|}^2}{z-{\lambda }_n}+{\int }_{{𝝈}_c}{\left|\widehat{f}(\lambda )\right|}^2\frac{d\lambda }{z-\lambda },$$ $f\in L^2\left(X\right)$, $z\notin 𝝈$.

ⓘ

Symbols:

$L^2(X,dx)$: Lebesgue–Stieltjes measurable, square integrable, complex-valued functions, $dx$: differential of $x$ , $\in$: element of, $⟨f,g⟩$: inner product over functions, $\int$: integral, $\notin$: not an element of, $z$: variable, $n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Referenced by:

§1.18(viii), §1.18(viii)

Permalink:

http://dlmf.nist.gov/1.18.E66

Encodings:

pMML, png, TeX

See also:

Annotations for §1.18(viii), §1.18 and Ch.1

…

18: 26.18 Counting Techniques

… ►

26.18.1

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Symbols:

$\left|A\right|$: number of elements of a finite set $A$ , $\cap$: intersection, $\setminus$: set subtraction, $\cup$: union, $j$: nonnegative integer, $n$: nonnegative integer, $A_k$: subsets and $S$: set

Permalink:

http://dlmf.nist.gov/26.18.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §26.18 and Ch.26

… ►

26.18.2

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Symbols:

$\lfloor x\rfloor$: floor of $x$ , $j$: nonnegative integer, $n$: nonnegative integer, $N$: set and $p_n$: primes

Permalink:

http://dlmf.nist.gov/26.18.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §26.18, §26.18 and Ch.26

… ►With the notation of §26.15, the number of placements of $n$ nonattacking rooks on an $n\times n$ chessboard that avoid the squares in a specified subset $B$ is ►

26.18.3 $$n!+\sum _{t=1}^n{(-1)}^t{r}_t(B)(n-t)!.$$

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Symbols:

$!$: factorial (as in $n!$), $n$: nonnegative integer, $r_j(B)$: number of rook positions and $B$: subset

Permalink:

http://dlmf.nist.gov/26.18.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §26.18, §26.18 and Ch.26

… ►

26.18.4 $$k^n+\sum _{t=1}^n{(-1)}^t\left(\genfrac{}{}{0.0pt}{}{k}{t}\right){(k-t)}^n.$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.18.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §26.18, §26.18 and Ch.26

…

19: 28.30 Expansions in Series of Eigenfunctions

… ►Then every continuous $2\pi$-periodic function $f(x)$ whose second derivative is square-integrable over the interval $[0,2\pi ]$ can be expanded in a uniformly and absolutely convergent series ►

28.30.3 $$f(x)=\sum _{m=0}^{\mathrm{\infty }}{f}_m{w}_m(x),$$

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Symbols:

$m$: integer, $x$: real variable and $w(z)$: Mathieu’s equation solution

Permalink:

http://dlmf.nist.gov/28.30.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.30(i), §28.30 and Ch.28

…

20: 22.11 Fourier and Hyperbolic Series

… ►

22.11.13 $${\mathrm{sn}}^2(z,k)=\frac{1}{k^2}\left(1-\frac{E}{K}\right)-\frac{2{\pi }^2}{k^2{K}^2}\sum _{n=1}^{\mathrm{\infty }}\frac{n{q}^n}{1-q^{2n}}\cos \left(2n\zeta \right).$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\cos z$: cosine function, $q$: nome, $z$: complex, $k$: modulus and $\zeta$: change of variable

Referenced by:

§22.11

Permalink:

http://dlmf.nist.gov/22.11.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §22.11 and Ch.22

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