Bateman-type%20sums

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11: 8 Incomplete Gamma and Related
Functions

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12: 28 Mathieu Functions and Hill’s Equation

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13: 26.2 Basic Definitions

… ►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. … ►

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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3	3	20	627	37	21637
…

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14: 8.26 Tables

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Khamis (1965) tabulates $P(a,x)$ for $a=0.05(.05)10(.1)20(.25)70$, $0.0001\le x\le 250$ to 10D.

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Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_n\left(x\right)$ for $n=2,3,4,10,20$, $x=0(.01)2$ to 7D; also $(x+n){\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2,3,4,10,20$, $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

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Pagurova (1961) tabulates $E_n\left(x\right)$ for $n=0(1)20$, $x=0(.01)2(.1)10$ to 4-9S; ${\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2(1)10$, $x=10(.1)20$ to 7D; ${\mathrm{e}}^x{E}_p\left(x\right)$ for $p=0(.1)1$, $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

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Zhang and Jin (1996, Table 19.1) tabulates $E_n\left(x\right)$ for $n=1,2,3,5,10,15,20$, $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

15: 23 Weierstrass Elliptic and Modular
Functions

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16: 4.41 Sums

§4.41 Sums

►For sums of hyperbolic functions see Gradshteyn and Ryzhik (2000, Chapter 1), Hansen (1975, §43), Prudnikov et al. (1986a, §5.3), and Zucker (1979).

17: 7.15 Sums

§7.15 Sums

►For sums involving the error function see Hansen (1975, p. 423) and Prudnikov et al. (1986b, vol. 2, pp. 650–651).

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

ⓘ

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

ⓘ

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

ⓘ

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

… ►

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

ⓘ

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

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

ⓘ

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

ⓘ

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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