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21: 24.17 Mathematical Applications

… ►Let $0\le h\le 1$ and $a,m$, and $n$ be integers such that $n>a$, $m>0$, and $f^{(m)}(x)$ is absolutely integrable over $[a,n]$. … ►

24.17.1 $$\sum _{j=a}^{n-1}{(-1)}^{j}f(j+h)=\frac{1}{2}\sum _{k=0}^{m-1}\frac{E_k\left(h\right)}{k!}\left({(-1)}^{n-1}{f}^{(k)}(n)+{(-1)}^a{f}^{(k)}(a)\right)+R_m(n),$$

ⓘ

Symbols:

$E_n\left(x\right)$: Euler polynomials, $!$: factorial (as in $n!$), $j$: integer, $k$: integer, $m$: integer, $h$: parameter, $a$: integer, $n$: integer, $f^{(m)}(x)$: integrable function and $R_m(n)$: remainder

Permalink:

http://dlmf.nist.gov/24.17.E1

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See also:

Annotations for §24.17(i), §24.17(i), §24.17 and Ch.24

… ►Let ${𝒮}_n$ denote the class of functions that have $n-1$ continuous derivatives on $ℝ$ and are polynomials of degree at most $n$ in each interval $(k,k+1)$, $k\in \mathbb{Z}$. … ►

24.17.4 $$S_n(k)={(-1)}^k,$$ $k\in \mathbb{Z}$.

ⓘ

Symbols:

$\in$: element of, $\mathbb{Z}$: set of all integers, $k$: integer and $n$: integer

Permalink:

http://dlmf.nist.gov/24.17.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §24.17(ii), §24.17(ii), §24.17 and Ch.24

… ►

24.17.6 $$M_n(k)=0,$$ $k\in \mathbb{Z}$.

ⓘ

Symbols:

$\in$: element of, $\mathbb{Z}$: set of all integers, $k$: integer, $n$: integer and $M_n(x)$: function

Referenced by:

§24.17(ii)

Permalink:

http://dlmf.nist.gov/24.17.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §24.17(ii), §24.17(ii), §24.17 and Ch.24

…

22: 24.4 Basic Properties

… ►

24.4.7 $$\sum _{k=1}^m{k}^n=\frac{B_{n+1}\left(m+1\right)-B_{n+1}}{n+1},$$

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $B_n\left(x\right)$: Bernoulli polynomials, $k$: integer, $m$: integer and $n$: integer

A&S Ref:

23.1.4

Referenced by:

§24.17(iii)

Permalink:

http://dlmf.nist.gov/24.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §24.4(iii), §24.4 and Ch.24

►

24.4.8 $$\sum _{k=1}^m{(-1)}^{m-k}{k}^n=\frac{E_n\left(m+1\right)+{(-1)}^m{E}_n\left(0\right)}{2}.$$

ⓘ

Symbols:

$E_n\left(x\right)$: Euler polynomials, $k$: integer, $m$: integer and $n$: integer

A&S Ref:

23.1.4

Referenced by:

§24.17(iii)

Permalink:

http://dlmf.nist.gov/24.4.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §24.4(iii), §24.4 and Ch.24

►

24.4.9 $$\sum _{k=0}^{m-1}{(a+dk)}^n=\frac{d^n}{n+1}\left(B_{n+1}\left(m+\frac{a}{d}\right)-B_{n+1}\left(\frac{a}{d}\right)\right),$$

ⓘ

Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $k$: integer, $m$: integer and $n$: integer

Referenced by:

§24.4(iii)

Permalink:

http://dlmf.nist.gov/24.4.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §24.4(iii), §24.4 and Ch.24

… ►

24.4.18 $$B_n\left(mx\right)=m^{n-1}\sum _{k=0}^{m-1}{B}_n\left(x+\frac{k}{m}\right).$$

ⓘ

Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $k$: integer, $m$: integer, $n$: integer and $x$: real or complex

A&S Ref:

23.1.10

Permalink:

http://dlmf.nist.gov/24.4.E18

Encodings:

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See also:

Annotations for §24.4(v), §24.4(v), §24.4 and Ch.24

… ►

24.4.20 $$E_n\left(mx\right)=m^n\sum _{k=0}^{m-1}{(-1)}^k{E}_n\left(x+\frac{k}{m}\right),$$ $m=1,3,5,\mathrm{\dots }$.

ⓘ

Symbols:

$E_n\left(x\right)$: Euler polynomials, $k$: integer, $m$: integer, $n$: integer and $x$: real or complex

A&S Ref:

23.1.10

Permalink:

http://dlmf.nist.gov/24.4.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §24.4(v), §24.4(v), §24.4 and Ch.24

…

23: 26.11 Integer Partitions: Compositions

§26.11 Integer Partitions: Compositions

►A composition is an integer partition in which order is taken into account. …The integer 0 is considered to have one composition consisting of no parts: … ►

26.11.3 $$c_m\left(n\right)=\left(\genfrac{}{}{0.0pt}{}{n-1}{m-1}\right),$$

ⓘ

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $c_m\left(n\right)$: number of compositions of $n$ into exactly $m$ parts, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §26.11 and Ch.26

►

26.11.4 $$\sum _{n=0}^{\mathrm{\infty }}{c}_m\left(n\right)q^n=\frac{q^m}{{(1-q)}^m}.$$

ⓘ

Symbols:

$c_m\left(n\right)$: number of compositions of $n$ into exactly $m$ parts, $m$: nonnegative integer, $n$: nonnegative integer and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.11.E4

Encodings:

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See also:

Annotations for §26.11 and Ch.26

…

24: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

… ►These are given by the following equations in which $a_1,a_2,\mathrm{\dots },a_n$ are nonnegative integers such that … ►

26.4.4 $$m=a_1+a_2+\mathrm{\cdots }+a_n.$$

ⓘ

Symbols:

$m$: nonnegative integer, $n$: nonnegative integer and $a_n$: nonnegative integers

Permalink:

http://dlmf.nist.gov/26.4.E4

Encodings:

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Annotations for §26.4(i), §26.4 and Ch.26

► $\lambda$ is a partition of $n$: ►

26.4.5 $$\lambda =1^{a_1},2^{a_2},\mathrm{\dots },n^{a_n}.$$

ⓘ

Symbols:

$n$: nonnegative integer, $a_n$: nonnegative integers and $\lambda$: integer partition

Permalink:

http://dlmf.nist.gov/26.4.E5

Encodings:

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Annotations for §26.4(i), §26.4 and Ch.26

… ►where the summation is over all nonnegative integers $n_1,n_2,\mathrm{\dots },n_k$ such that $n_1+n_2+\mathrm{\cdots }+n_k=n$. …

25: 24.5 Recurrence Relations

… ►

24.5.1 $$\sum _{k=0}^{n-1}\left(\genfrac{}{}{0pt}{}{n}{k}\right)B_k\left(x\right)=n{x}^{n-1},$$ $n=2,3,\mathrm{\dots }$,

ⓘ

Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer, $n$: integer and $x$: real or complex

Permalink:

http://dlmf.nist.gov/24.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §24.5(i), §24.5 and Ch.24

►

24.5.2 $$\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)E_k\left(x\right)+E_n\left(x\right)=2x^n,$$ $n=1,2,\mathrm{\dots }$.

ⓘ

Symbols:

$E_n\left(x\right)$: Euler polynomials, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer, $n$: integer and $x$: real or complex

Permalink:

http://dlmf.nist.gov/24.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §24.5(i), §24.5 and Ch.24

►

24.5.3 $$\sum _{k=0}^{n-1}\left(\genfrac{}{}{0pt}{}{n}{k}\right)B_k=0,$$ $n=2,3,\mathrm{\dots }$,

ⓘ

Symbols:

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

Referenced by:

§24.19(i)

Permalink:

http://dlmf.nist.gov/24.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §24.5(i), §24.5 and Ch.24

►

24.5.4 $$\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{2n}{2k}\right)E_{2k}=0,$$ $n=1,2,\mathrm{\dots }$,

ⓘ

Symbols:

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

Referenced by:

§24.19(i)

Permalink:

http://dlmf.nist.gov/24.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §24.5(i), §24.5 and Ch.24

►

24.5.5 $$\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)2^k{E}_{n-k}+E_n=2.$$

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/24.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §24.5(i), §24.5 and Ch.24

…

26: 26.2 Basic Definitions

… ►If the set consists of the integers 1 through $n$, a permutation $\sigma$ can be thought of as a rearrangement of these integers where the integer in position $j$ is $\sigma (j)$. … ►If, for example, a permutation of the integers 1 through 6 is denoted by $256413$, then the cycles are $\left(1,2,5\right)$, $\left(3,6\right)$, and $\left(4\right)$. … ►A lattice path is a directed path on the plane integer lattice $\{0,1,2,\mathrm{\dots }\}\times \{0,1,2,\mathrm{\dots }\}$. … ►A partition of a nonnegative integer $n$ is an unordered collection of positive integers whose sum is $n$. … ►The integers whose sum is $n$ are referred to as the parts in the partition. …

27: 34.1 Special Notation

… ► ►►►

$2j_1,2j_2,2j_3,2l_1,2l_2,2l_3$	nonnegative integers.
$r,s,t$	nonnegative integers.

… ►

34.1.1

ⓘ

Symbols:

: $\mathit{3}j$ symbol and $j,j_r$: non-negative integers or non-negative integers plus one half.

Sources:

Edmonds (1974, p. 46, Eq. (3.7.3)); Rotenberg et al. (1959, p. 1, Eq. (1.1a))

Permalink:

http://dlmf.nist.gov/34.1.E1

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.15):

A wording change reflects that the Clebsch–Gordan coefficients are an alternative formulation of angular momentum problems, rather than alternative notation for the $\mathit{3}j$. The sign of $m_3$ was changed for clarity.

See also:

Annotations for §34.1 and Ch.34

…

28: 1.1 Special Notation

… ► ►►►►

$x,y$	real variables.
…
$j,k,\mathrm{\ell }$	integers.
$m,n$	nonnegative integers, unless specified otherwise.
…

…

29: 4.22 Infinite Products and Partial Fractions

… ►

4.22.1 $$\sin z=z\prod _{n=1}^{\mathrm{\infty }}\left(1-\frac{z^2}{n^2{\pi }^2}\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.89

Permalink:

http://dlmf.nist.gov/4.22.E1

Encodings:

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See also:

Annotations for §4.22 and Ch.4

►

4.22.2 $$\cos z=\prod _{n=1}^{\mathrm{\infty }}\left(1-\frac{4z^2}{{(2n-1)}^2{\pi }^2}\right).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.90

Permalink:

http://dlmf.nist.gov/4.22.E2

Encodings:

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See also:

Annotations for §4.22 and Ch.4

►When $z\ne n\pi$, $n\in \mathbb{Z}$, ►

4.22.3 $$\cot z=\frac{1}{z}+2z\sum _{n=1}^{\mathrm{\infty }}\frac{1}{z^2-n^2{\pi }^2},$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.91

Referenced by:

§22.12

Permalink:

http://dlmf.nist.gov/4.22.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.22 and Ch.4

►

4.22.4 $${\csc }^{2}z=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{1}{{(z-n\pi )}^2},$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.92

Permalink:

http://dlmf.nist.gov/4.22.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.22 and Ch.4

…

30: 4.36 Infinite Products and Partial Fractions

… ►

4.36.1 $$\sinh z=z\prod _{n=1}^{\mathrm{\infty }}\left(1+\frac{z^2}{n^2{\pi }^2}\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\sinh z$: hyperbolic sine function, $n$: integer and $z$: complex variable

A&S Ref:

4.5.68

Permalink:

http://dlmf.nist.gov/4.36.E1

Encodings:

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See also:

Annotations for §4.36 and Ch.4

►

4.36.2 $$\cosh z=\prod _{n=1}^{\mathrm{\infty }}\left(1+\frac{4z^2}{{(2n-1)}^2{\pi }^2}\right).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cosh z$: hyperbolic cosine function, $n$: integer and $z$: complex variable

A&S Ref:

4.5.69

Permalink:

http://dlmf.nist.gov/4.36.E2

Encodings:

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See also:

Annotations for §4.36 and Ch.4

►When $z\ne n\pi \mathrm{i}$, $n\in \mathbb{Z}$, ►

4.36.3 $$\coth z=\frac{1}{z}+2z\sum _{n=1}^{\mathrm{\infty }}\frac{1}{z^2+n^2{\pi }^2},$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\coth z$: hyperbolic cotangent function, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.36.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.36 and Ch.4

►

4.36.4 $${\mathrm{csch}}^{2}z=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{1}{{(z-n\pi \mathrm{i})}^2},$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{csch}z$: hyperbolic cosecant function, $\mathrm{i}$: imaginary unit, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.36.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.36 and Ch.4

…