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1: 28.2 Definitions and Basic Properties

… ►The general solution of (28.2.16) is $\nu =\pm \widehat{\nu }+2n$, where $n\in \mathbb{Z}$. … ►

28.2.19 $$q{c}_{2n+2}-\left(a-{(\nu +2n)}^2\right)c_{2n}+q{c}_{2n-2}=0,$$ $n\in \mathbb{Z}$.

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Symbols:

$\in$: element of, $\mathbb{Z}$: set of all integers, $q=h^2$: parameter, $n$: integer, $\nu$: complex parameter, $a$: parameter and $c_{2n}$: coefficients

A&S Ref:

20.3.12 20.3.13

Referenced by:

§28.2(iv)

Permalink:

http://dlmf.nist.gov/28.2.E19

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Annotations for §28.2(iv), §28.2 and Ch.28

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28.2.23 $$a_n\left(0\right)=n^2,$$ $n=0,1,2,\mathrm{\dots }$,

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Symbols:

$a_n\left(q\right)$: eigenvalues of Mathieu equation and $n$: integer

Permalink:

http://dlmf.nist.gov/28.2.E23

Encodings:

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See also:

Annotations for §28.2(v), §28.2 and Ch.28

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28.2.24 $$b_n\left(0\right)=n^2,$$ $n=1,2,3,\mathrm{\dots }$.

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Symbols:

$b_n\left(q\right)$: eigenvalues of Mathieu equation and $n$: integer

Permalink:

http://dlmf.nist.gov/28.2.E24

Encodings:

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See also:

Annotations for §28.2(v), §28.2 and Ch.28

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§28.2(vi) Eigenfunctions

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2: 32.1 Special Notation

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$m,n$	integers.
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3: 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:

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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:

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See also:

Annotations for §24.6 and Ch.24

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24.6.3 $$B_{2n}=\sum _{k=1}^n\frac{(k-1)!k!}{(2k+1)!}\sum _{j=1}^k{(-1)}^{j-1}\left(\genfrac{}{}{0pt}{}{2k}{k+j}\right)j^{2n}.$$

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Symbols:

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

Referenced by:

§24.6

Permalink:

http://dlmf.nist.gov/24.6.E3

Encodings:

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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:

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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:

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See also:

Annotations for §24.6 and Ch.24

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4: 27.19 Methods of Computation: Factorization

§27.19 Methods of Computation: Factorization

►Techniques for factorization of integers fall into three general classes: Deterministic algorithms, Type I probabilistic algorithms whose expected running time depends on the size of the smallest prime factor, and Type II probabilistic algorithms whose expected running time depends on the size of the number to be factored. … ►Type II probabilistic algorithms for factoring $n$ rely on finding a pseudo-random pair of integers $(x,y)$ that satisfy $x^2\equiv y^2(modn)$. …As of January 2009 the snfs holds the record for the largest integer that has been factored by a Type II probabilistic algorithm, a 307-digit composite integer. …The largest composite numbers that have been factored by other Type II probabilistic algorithms are a 63-digit integer by cfrac, a 135-digit integer by mpqs, and a 182-digit integer by gnfs. …

5: 27.6 Divisor Sums

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27.6.1

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Symbols:

$\lambda \left(n\right)$: Liouville’s function, $d$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.6.E1

Encodings:

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Annotations for §27.6 and Ch.27

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27.6.2 $$\sum _{d|n}\mu \left(d\right)f(d)=\prod _{p|n}(1-f(p)),$$ $n>1$.

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Symbols:

$\mu \left(n\right)$: Möbius function, $d$: positive integer, $n$: positive integer and $p,p_1,\mathrm{\dots }$: prime numbers

Permalink:

http://dlmf.nist.gov/27.6.E2

Encodings:

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See also:

Annotations for §27.6 and Ch.27

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27.6.6 $$\sum _{d|n}{\varphi }_k\left(d\right){\left(\frac{n}{d}\right)}^k=1^k+2^k+\mathrm{\cdots }+n^k,$$

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Symbols:

${\varphi }_k\left(n\right)$: sum of powers of integers relatively prime to a number, $d$: positive integer, $k$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.6.E6

Encodings:

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See also:

Annotations for §27.6 and Ch.27

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27.6.7 $$\sum _{d|n}\mu \left(d\right){\left(\frac{n}{d}\right)}^k=J_k\left(n\right),$$

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Symbols:

$J_k\left(n\right)$: Jordan’s function, $\mu \left(n\right)$: Möbius function, $d$: positive integer, $k$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.6.E7

Encodings:

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See also:

Annotations for §27.6 and Ch.27

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27.6.8 $$\sum _{d|n}{J}_k\left(d\right)=n^k.$$

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Symbols:

$J_k\left(n\right)$: Jordan’s function, $d$: positive integer, $k$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.6.E8

Encodings:

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See also:

Annotations for §27.6 and Ch.27

6: 26.9 Integer Partitions: Restricted Number and Part Size

§26.9 Integer Partitions: Restricted Number and Part Size

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§26.9(i) Definitions

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Table 26.9.1: Partitions $p_k\left(n\right)$.

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$n$	$k$
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§26.9(ii) Generating Functions

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§26.9(iii) Recurrence Relations

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7: 14.27 Zeros

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, $\mu \notin \mathbb{Z}$, $\nu \in \mathbb{Z}$, and $\sin \left((\mu -\nu )\pi \right)$ and $\sin \left(\mu \pi \right)$ have opposite signs.

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$\mu ,\nu \in \mathbb{Z}$, , and $\nu$ is odd.

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8: 26.10 Integer Partitions: Other Restrictions

§26.10 Integer Partitions: Other Restrictions

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§26.10(i) Definitions

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Table 26.10.1: Partitions restricted by difference conditions, or equivalently with parts from $A_{j,k}$.

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	$p(𝒟,n)$	$p(𝒟2,n)$	$p(𝒟2,\in T,n)$	$p({𝒟}^{\prime }3,n)$
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§26.10(ii) Generating Functions

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9: 26.12 Plane Partitions

… ►A plane partition, $\pi$, of a positive integer $n$, is a partition of $n$ in which the parts have been arranged in a 2-dimensional array that is weakly decreasing (nonincreasing) across rows and down columns. … ►

26.12.3 $$B(r,s,t)=\{(h,j,k)|\mathrm{\hspace{0.25em}1}\le h\le r,1\le j\le s,1\le k\le t\}.$$

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Symbols:

$r$: positive integer, $s$: positive integer, $t$: positive integer, $h$: positive integer, $j$: positive integer, $k$: positive integer and $B(r,s,t)$: Box

Permalink:

http://dlmf.nist.gov/26.12.E3

Encodings:

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See also:

Annotations for §26.12(i), §26.12 and Ch.26

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26.12.9 $${\left(\prod _{h=1}^r\prod _{j=1}^s\frac{h+j+t-1}{h+j-1}\right)}^2;$$

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Symbols:

$r$: positive integer, $s$: positive integer, $t$: positive integer, $h$: positive integer and $j$: positive integer

Permalink:

http://dlmf.nist.gov/26.12.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §26.12(i), §26.12 and Ch.26

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26.12.13 $$\prod _{h=1}^r\prod _{j=1}^r\frac{h+j+t-1}{h+j-1};$$

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Symbols:

$r$: positive integer, $t$: positive integer, $h$: positive integer and $j$: positive integer

Permalink:

http://dlmf.nist.gov/26.12.E13

Encodings:

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See also:

Annotations for §26.12(i), §26.12 and Ch.26

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26.12.14 $$\prod _{h=1}^r\prod _{j=1}^{r+1}\frac{h+j+t-1}{h+j-1}.$$

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Symbols:

$r$: positive integer, $t$: positive integer, $h$: positive integer and $j$: positive integer

Permalink:

http://dlmf.nist.gov/26.12.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §26.12(i), §26.12 and Ch.26

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10: 24.10 Arithmetic Properties

… ►where $n\ge 2$, and $\mathrm{\ell }(\ge 1)$ is an arbitrary integer such that $(p-1)p^{\mathrm{\ell }}|2n$. … ►valid when $m\equiv n(mod(p-1)p^{\mathrm{\ell }})$ and $n\cancel{\equiv }0(modp-1)$, where $\mathrm{\ell }(\ge 0)$ is a fixed integer. …valid for fixed integers $\mathrm{\ell }(\ge 0)$, and for all $n(\ge 0)$ and $w(\ge 0)$ such that $2^{\mathrm{\ell }}|w$. … ►where $M(\ge 2)$ and $b$ are integers, with $b$ relatively prime to $M$. … ►valid for fixed integers $\mathrm{\ell }(\ge 1)$, and for all $n(\ge 1)$ such that $2n\cancel{\equiv }0$ $(modp-1)$ and $p^{\mathrm{\ell }}|2n$. …