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1: 34.13 Methods of Computation

… ►Methods of computation for $\mathit{3}j$ and $\mathit{6}j$ symbols include recursion relations, see Schulten and Gordon (1975a), Luscombe and Luban (1998), and Edmonds (1974, pp. 42–45, 48–51, 97–99); summation of single-sum expressions for these symbols, see Varshalovich et al. (1988, §§8.2.6, 9.2.1) and Fang and Shriner (1992); evaluation of the generalized hypergeometric functions of unit argument that represent these symbols, see Srinivasa Rao and Venkatesh (1978) and Srinivasa Rao (1981). …

2: 26.3 Lattice Paths: Binomial Coefficients

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Table 26.3.1: Binomial coefficients $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$.

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$m$	$n$
	…
10	1	10	45	120	210	252	210	120	45	10	1

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Table 26.3.2: Binomial coefficients $\left(\genfrac{}{}{0.0pt}{}{m+n}{m}\right)$ for lattice paths.

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$m$	$n$
	…
2	1	3	6	10	15	21	28	36	45
…
8	1	9	45	165	495	1287	3003	6435	12870

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…

3: 26.16 Multiset Permutations

… ►Additional information can be found in Andrews (1976, pp. 39–45). …

4: 20.15 Tables

… ►Theta functions are tabulated in Jahnke and Emde (1945, p. 45). …

5: 26.2 Basic Definitions

… ►

Table 26.2.1: Partitions $p\left(n\right)$.

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$n$	$p\left(n\right)$	$n$	$p\left(n\right)$	$n$	$p\left(n\right)$
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11	56	28	3718	45	89134
…

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6: 4.19 Maclaurin Series and Laurent Series

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4.19.6 $$\cot z=\frac{1}{z}-\frac{z}{3}-\frac{z^3}{45}-\frac{2}{945}{z}^5-\mathrm{\cdots }-\frac{{(-1)}^{n-1}{2}^{2n}{B}_{2n}}{(2n)!}{z}^{2n-1}-\mathrm{\cdots },$$ ,

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Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $!$: factorial (as in $n!$), $n$: integer and $z$: complex variable

A&S Ref:

4.3.70

Permalink:

http://dlmf.nist.gov/4.19.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.19 and Ch.4

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7: 5.7 Series Expansions

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5.7.1 $$\frac{1}{\mathrm{\Gamma }\left(z\right)}=\sum _{k=1}^{\mathrm{\infty }}{c}_k{z}^k,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $k$: nonnegative integer, $z$: complex variable and $c_k$: coefficient

A&S Ref:

6.1.34

Referenced by:

§5.7(i)

Permalink:

http://dlmf.nist.gov/5.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §5.7(i), §5.7 and Ch.5

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5.7.5 $$\psi \left(1+z\right)=\frac{1}{2z}-\frac{\pi }{2}\cot \left(\pi z\right)+\frac{1}{z^2-1}+1-\gamma -\sum _{k=1}^{\mathrm{\infty }}(\zeta \left(2k+1\right)-1)z^{2k},$$ , $z\ne 0,\pm 1$.

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Symbols:

$\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\psi \left(z\right)$: psi (or digamma) function, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

6.3.15

Referenced by:

§5.7(i)

Permalink:

http://dlmf.nist.gov/5.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §5.7(i), §5.7 and Ch.5

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5.7.6 $$\psi \left(z\right)=-\gamma -\frac{1}{z}+\sum _{k=1}^{\mathrm{\infty }}\frac{z}{k(k+z)}=-\gamma +\sum _{k=0}^{\mathrm{\infty }}\left(\frac{1}{k+1}-\frac{1}{k+z}\right),$$

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Symbols:

$\gamma$: Euler’s constant, $\psi \left(z\right)$: psi (or digamma) function, $k$: nonnegative integer and $z$: complex variable

Referenced by:

§5.19(i)

Permalink:

http://dlmf.nist.gov/5.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §5.7(ii), §5.7 and Ch.5

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5.7.7 $$\psi \left(\frac{z+1}{2}\right)-\psi \left(\frac{z}{2}\right)=2\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k}{k+z}.$$

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Symbols:

$\psi \left(z\right)$: psi (or digamma) function, $k$: nonnegative integer and $z$: complex variable

Referenced by:

§15.4(iii)

Permalink:

http://dlmf.nist.gov/5.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §5.7(ii), §5.7 and Ch.5

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5.7.8 $$\mathrm{\Im }\psi \left(1+\mathrm{i}y\right)=\sum _{k=1}^{\mathrm{\infty }}\frac{y}{k^2+y^2}.$$

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Symbols:

$\psi \left(z\right)$: psi (or digamma) function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer and $y$: real variable

A&S Ref:

6.3.13

Permalink:

http://dlmf.nist.gov/5.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §5.7(ii), §5.7 and Ch.5

8: 24.6 Explicit Formulas

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24.6.1 $$B_{2n}=\sum _{k=2}^{2n+1}\frac{{(-1)}^{k-1}}{k}\left(\genfrac{}{}{0pt}{}{2n+1}{k}\right)\sum _{j=1}^{k-1}{j}^{2n},$$

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Symbols:

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

Referenced by:

§24.6

Permalink:

http://dlmf.nist.gov/24.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §24.6 and Ch.24

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24.6.2 $$B_n=\frac{1}{n+1}\sum _{k=1}^n\sum _{j=1}^k{(-1)}^j{j}^n\left(\genfrac{}{}{0.0pt}{}{n+1}{k-j}\right)/\left(\genfrac{}{}{0.0pt}{}{n}{k}\right),$$

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Symbols:

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

Referenced by:

§24.6

Permalink:

http://dlmf.nist.gov/24.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §24.6 and Ch.24

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24.6.4 $$E_{2n}=\sum _{k=1}^n\frac{1}{2^{k-1}}\sum _{j=1}^k{(-1)}^j\left(\genfrac{}{}{0pt}{}{2k}{k-j}\right)j^{2n},$$

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Symbols:

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

Referenced by:

§24.6

Permalink:

http://dlmf.nist.gov/24.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §24.6 and Ch.24

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24.6.6 $$E_{2n}=\sum _{k=1}^{2n}\frac{{(-1)}^k}{2^{k-1}}\left(\genfrac{}{}{0pt}{}{2n+1}{k+1}\right)\sum _{j=0}^{\lfloor \frac{1}{2}k-\frac{1}{2}\rfloor }\left(\genfrac{}{}{0pt}{}{k}{j}\right){(k-2j)}^{2n}.$$

ⓘ

Symbols:

$E_n$: Euler numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\lfloor x\rfloor$: floor of $x$ , $j$: integer, $k$: integer and $n$: integer

Referenced by:

§24.6

Permalink:

http://dlmf.nist.gov/24.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §24.6 and Ch.24

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24.6.9 $$B_n=\sum _{k=0}^n\frac{1}{k+1}\sum _{j=0}^k{(-1)}^j\left(\genfrac{}{}{0pt}{}{k}{j}\right)j^n,$$

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Symbols:

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

Referenced by:

§24.6

Permalink:

http://dlmf.nist.gov/24.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §24.6 and Ch.24

…

9: 25.8 Sums

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25.8.5 $$\sum _{k=2}^{\mathrm{\infty }}\zeta \left(k\right)z^k=-\gamma z-z\psi \left(1-z\right),$$ .

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Symbols:

$\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, $\psi \left(z\right)$: psi (or digamma) function, $k$: nonnegative integer and $z$: complex variable

Keywords:

infinite series

Source:

Erdélyi et al. (1953a, (1.17.5), p. 45); with $z\mapsto -z$

Referenced by:

(25.8.7)

Permalink:

http://dlmf.nist.gov/25.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §25.8 and Ch.25

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10: 3.9 Acceleration of Convergence

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Table 3.9.1: Shanks’ transformation for $s_n={\sum }_{j=1}^n{(-1)}^{j+1}{j}^{-2}$.

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$n$	$t_{n,2}$	$t_{n,4}$	$t_{n,6}$	$t_{n,8}$	$t_{n,10}$
…
1	0.82692 30769 23	0.82259 02017 65	0.82247 05346 57	0.82246 71342 06	0.82246 70363 45
2	0.82111 11111 11	0.82243 44785 14	0.82246 61821 45	0.82246 70102 48	0.82246 70327 79
…

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