Bateman-type%20sums

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21: 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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22: 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 $$\zeta \left(3\right)=\frac{5}{2}\sum _{k=1}^{\mathrm{\infty }}\frac{{(-1)}^{k-1}}{k^3\left(\genfrac{}{}{0.0pt}{}{2k}{k}\right)}.$$

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

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

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

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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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25: 36 Integrals with Coalescing Saddles

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26: 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. …

27: 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. …

28: 33.24 Tables

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Abramowitz and Stegun (1964, Chapter 14) tabulates $F_0(\eta ,\rho )$, $G_0(\eta ,\rho )$, $F_0^{\prime }(\eta ,\rho )$, and $G_0^{\prime }(\eta ,\rho )$ for $\eta =0.5(.5)20$ and $\rho =1(1)20$, 5S; $C_0\left(\eta \right)$ for $\eta =0(.05)3$, 6S.

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29: 26.9 Integer Partitions: Restricted Number and Part Size

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26.9.5 $$\sum _{n=0}^{\mathrm{\infty }}{p}_k\left(n\right)q^n=\prod _{j=1}^k\frac{1}{1-q^j}=1+\sum _{m=1}^{\mathrm{\infty }}{\left[\genfrac{}{}{0.0pt}{}{k+m-1}{m}\right]}_q{q}^m,$$

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Symbols:

$p_k\left(n\right)$: total number of partitions of $n$ into at most $k$ parts, ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $j$: nonnegative integer, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.9.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §26.9(ii), §26.9 and Ch.26

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26.9.6 $$\sum _{n=0}^{\mathrm{\infty }}{p}_k(\le m,n)q^n={\left[\genfrac{}{}{0.0pt}{}{m+k}{k}\right]}_q.$$

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Symbols:

${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $p_k(\le m,n)$: number of partitions of $n$ into at most $k$ parts, each less than or equal to $m$ , $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.9.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §26.9(ii), §26.9 and Ch.26

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26.9.7 $$\sum _{m,n=0}^{\mathrm{\infty }}{p}_k(\le m,n)x^k{q}^n=1+\sum _{k=1}^{\mathrm{\infty }}{\left[\genfrac{}{}{0.0pt}{}{m+k}{k}\right]}_q{x}^k=\prod _{j=0}^m\frac{1}{1-x{q}^j}.$$

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Symbols:

${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $p_k(\le m,n)$: number of partitions of $n$ into at most $k$ parts, each less than or equal to $m$ , $x$: real variable, $j$: nonnegative integer, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.9.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §26.9(ii), §26.9 and Ch.26

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26.9.9 $$p_k\left(n\right)=\frac{1}{n}\sum _{t=1}^n{p}_k\left(n-t\right)\sum _{\begin{array}{c}j|t\\ j\le k\end{array}}j,$$

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Symbols:

$p_k\left(n\right)$: total number of partitions of $n$ into at most $k$ parts, $j$: nonnegative integer, $k$: nonnegative integer and $n$: nonnegative integer

Referenced by:

§26.9(iii)

Permalink:

http://dlmf.nist.gov/26.9.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §26.9(iii), §26.9 and Ch.26

►where the inner sum is taken over all positive divisors of $t$ that are less than or equal to $k$. …

30: 27.15 Chinese Remainder Theorem

… ►Their product $m$ has 20 digits, twice the number of digits in the data. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result $(modm)$, which is correct to 20 digits. …