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21: 20.11 Generalizations and Analogs

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§20.11(i) Gauss Sum

►For relatively prime integers $m,n$ with $n>0$ and $mn$ even, the Gauss sum $G(m,n)$ is defined by ►

20.11.1 $$G(m,n)=\sum _{k=0}^{n-1}{\mathrm{e}}^{-\pi \mathrm{i}{k}^{2}m/n};$$

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Defines:

$G(m,n)$: Gauss sum (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $m$: integer and $n$: integer

Permalink:

http://dlmf.nist.gov/20.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §20.11(i), §20.11 and Ch.20

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20.11.3 $$f(a,b)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{a}^{n(n+1)/2}{b}^{n(n-1)/2},$$

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Symbols:

$n$: integer

Permalink:

http://dlmf.nist.gov/20.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §20.11(ii), §20.11 and Ch.20

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22: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

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Table 26.4.1: Multinomials and partitions.

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$n$	$m$	$\lambda$	$M_1$	$M_2$	$M_3$
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$5$	$2$	$2^1,3^1$	$10$	$20$	$10$
$5$	$3$	$1^2,3^1$	$20$	$20$	$10$
…

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26.4.9 $${(x_1+x_2+\mathrm{\cdots }+x_k)}^n=\sum \left(\genfrac{}{}{0.0pt}{}{n}{n_1,n_2,\mathrm{\dots },n_k}\right)x_1^{n_1}{x}_2^{n_2}\mathrm{\cdots }{x}_k^{n_k},$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{n_1+n_2+\mathrm{\cdots }+n_k}{n_1,n_2,\mathrm{\dots },n_k}\right)$: multinomial coefficient, $x$: real variable, $k$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

24.1.2

Permalink:

http://dlmf.nist.gov/26.4.E9

Encodings:

pMML, png, TeX

See also:

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

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26.4.10 $$\left(\genfrac{}{}{0.0pt}{}{n_1+n_2+\mathrm{\cdots }+n_m}{n_1,n_2,\mathrm{\dots },n_m}\right)=\sum _{k=1}^m\left(\genfrac{}{}{0.0pt}{}{n_1+n_2+\mathrm{\cdots }+n_m-1}{n_1,n_2,\mathrm{\dots },n_{k-1},n_k-1,n_{k+1},\mathrm{\dots },n_m}\right),$$ $n_1,n_2,\mathrm{\dots },n_m\ge 1$.

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{n_1+n_2+\mathrm{\cdots }+n_k}{n_1,n_2,\mathrm{\dots },n_k}\right)$: multinomial coefficient, $k$: nonnegative integer, $m$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

24.1.2

Permalink:

http://dlmf.nist.gov/26.4.E10

Encodings:

pMML, png, TeX

See also:

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

23: 26.5 Lattice Paths: Catalan Numbers

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Table 26.5.1: Catalan numbers.

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$n$	$C\left(n\right)$	$n$	$C\left(n\right)$	$n$	$C\left(n\right)$
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6	132	13	7 42900	20	65641 20420

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26.5.2 $$\sum _{n=0}^{\mathrm{\infty }}C\left(n\right)x^n=\frac{1-\sqrt{1-4x}}{2x},$$ .

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Symbols:

$C\left(n\right)$: Catalan number, $x$: real variable and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §26.5(ii), §26.5 and Ch.26

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26.5.3 $$C\left(n+1\right)=\sum _{k=0}^{n}C\left(k\right)C\left(n-k\right),$$

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Symbols:

$C\left(n\right)$: Catalan number, $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §26.5(iii), §26.5 and Ch.26

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26.5.5 $$C\left(n+1\right)=\sum _{k=0}^{\lfloor n/2\rfloor }\left(\genfrac{}{}{0.0pt}{}{n}{2k}\right)2^{n-2k}C\left(k\right).$$

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Symbols:

$C\left(n\right)$: Catalan number, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\lfloor x\rfloor$: floor of $x$ , $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §26.5(iii), §26.5 and Ch.26

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24: 3.4 Differentiation

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$$B_{-2}^5=\frac{1}{120}(6-10t-15t^2+20t^3-5t^4),$$

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$$B_{-3}^6=-\frac{1}{720}(12-8t-45t^2+20t^3+15t^4-6t^5),$$

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$$B_{-2}^6=\frac{1}{60}(9-9t-30t^2+20t^3+5t^4-3t^5),$$

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$$B_2^6=-\frac{1}{60}(9+9t-30t^2-20t^3+5t^4+3t^5),$$

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$$B_3^6=\frac{1}{720}(12+8t-45t^2-20t^3+15t^4+6t^5).$$

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25: 25.6 Integer Arguments

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25.6.3 $$\zeta \left(-n\right)=-\frac{B_{n+1}}{n+1},$$ $n=1,2,3,\mathrm{\dots }$.

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Symbols:

$B_n$: Bernoulli numbers, $\zeta \left(s\right)$: Riemann zeta function and $n$: nonnegative integer

Source:

Apostol (1976, (20), p. 266); with $n\ne 0$

A&S Ref:

23.2.15 (in slightly different form)

Permalink:

http://dlmf.nist.gov/25.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.6(i), §25.6 and Ch.25

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25.6.5 $$\zeta \left(k+1\right)=\frac{1}{k!}\sum _{n_1=1}^{\mathrm{\infty }}\mathrm{\dots }\sum _{n_k=1}^{\mathrm{\infty }}\frac{1}{n_1\mathrm{\cdots }{n}_k(n_1+\mathrm{\cdots }+n_k)},$$ $k=1,2,3,\mathrm{\dots }$.

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $!$: factorial (as in $n!$), $k$: nonnegative integer and $n$: nonnegative integer

Source:

Mordell (1958, (5), p. 369)

Permalink:

http://dlmf.nist.gov/25.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §25.6(i), §25.6 and Ch.25

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25.6.8 $$\zeta \left(2\right)=3\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k^2\left(\genfrac{}{}{0.0pt}{}{2k}{k}\right)}.$$

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Symbols:

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

Keywords:

infinite series

Source:

van der Poorten (1980, (2), p. 271)

Permalink:

http://dlmf.nist.gov/25.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §25.6(i), §25.6 and Ch.25

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25.6.9 $$\zeta \left(3\right)=\frac{5}{2}\sum _{k=1}^{\mathrm{\infty }}\frac{{(-1)}^{k-1}}{k^3\left(\genfrac{}{}{0.0pt}{}{2k}{k}\right)}.$$

ⓘ

Symbols:

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

Keywords:

infinite series

Source:

van der Poorten (1980, (3), p. 271)

Permalink:

http://dlmf.nist.gov/25.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §25.6(i), §25.6 and Ch.25

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25.6.19 $$\left(m+n+\frac{3}{2}\right)\zeta \left(2m+2n+2\right)=\left(\sum _{k=1}^m+\sum _{k=1}^n\right)\zeta \left(2k\right)\zeta \left(2m+2n+2-2k\right),$$ $m\ge 0$, $n\ge 0$, $m+n\ge 1$.

ⓘ

Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $k$: nonnegative integer, $m$: nonnegative integer and $n$: nonnegative integer

Source:

Basu and Apostol (2000, (3.12), p. 404)

Permalink:

http://dlmf.nist.gov/25.6.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §25.6(iii), §25.6 and Ch.25

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26: 23.9 Laurent and Other Power Series

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23.9.1 $$c_n=(2n-1)\sum _{w\in 𝕃\setminus \{0\}}{w}^{-2n},$$ $n=2,3,4,\mathrm{\dots }$.

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Symbols:

$\in$: element of, $\setminus$: set subtraction, $𝕃$: lattice, $n$: integer and $c_n$

Referenced by:

§20.6

Permalink:

http://dlmf.nist.gov/23.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §23.9 and Ch.23

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23.9.2 $$\mathrm{\wp }\left(z\right)=\frac{1}{z^2}+\sum _{n=2}^{\mathrm{\infty }}{c}_n{z}^{2n-2},$$ ,

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $𝕃$: lattice, $n$: integer, $z$: complex and $c_n$

A&S Ref:

18.5.1

Referenced by:

§23.9, §23.9

Permalink:

http://dlmf.nist.gov/23.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §23.9 and Ch.23

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23.9.3 $$\zeta \left(z\right)=\frac{1}{z}-\sum _{n=2}^{\mathrm{\infty }}\frac{c_n}{2n-1}{z}^{2n-1},$$ .

ⓘ

Symbols:

$\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $𝕃$: lattice, $n$: integer, $z$: complex and $c_n$

A&S Ref:

18.5.5

Referenced by:

§23.9

Permalink:

http://dlmf.nist.gov/23.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §23.9 and Ch.23

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$c_2={\displaystyle \frac{1}{20}}{g}_2,$

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23.9.5 $$c_n=\frac{3}{(2n+1)(n-3)}\sum _{m=2}^{n-2}{c}_m{c}_{n-m},$$ $n\ge 4$.

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Symbols:

$n$: integer, $m$: integer and $c_n$

A&S Ref:

18.5.3

Referenced by:

§23.9

Permalink:

http://dlmf.nist.gov/23.9.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §23.9 and Ch.23

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27: 26.14 Permutations: Order Notation

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26.14.1

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Symbols:

$j$: nonnegative integer, $k$: nonnegative integer, $n$: nonnegative integer and $\sigma$: permutation

Permalink:

http://dlmf.nist.gov/26.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §26.14(i), §26.14 and Ch.26

►Equivalently, this is the sum over of the number of integers less than $\sigma (j)$ that lie in positions to the right of the $j$th position: $inv(35247816)=2+3+1+1+2+2+0=11.$ … ►The major index is the sum of all positions that mark the first element of a descent: ►

26.14.2

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Symbols:

$j$: nonnegative integer, $n$: nonnegative integer, $\sigma$: permutation and $maj(\sigma )$: major index

Permalink:

http://dlmf.nist.gov/26.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §26.14(i), §26.14 and Ch.26

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26.14.3 $$\sum _{\sigma \in {𝔖}_n}{q}^{inv(\sigma )}=\sum _{\sigma \in {𝔖}_n}{q}^{maj(\sigma )}=\prod _{j=1}^n\frac{1-q^j}{1-q}.$$

ⓘ

Symbols:

$\in$: element of, ${𝔖}_n$: set of permutations of $\{1,2,\mathrm{\dots },n\}$ , $j$: nonnegative integer, $n$: nonnegative integer and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §26.14(ii), §26.14 and Ch.26

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28: 36 Integrals with Coalescing Saddles

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29: Gergő Nemes

… ►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions. …

30: Wolter Groenevelt

… ►As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. …