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11: Mathematical Introduction

… ►The NIST Handbook has essentially the same objective as the Handbook of Mathematical Functions that was issued in 1964 by the National Bureau of Standards as Number 55 in the NBS Applied Mathematics Series (AMS). … ►Particular care is taken with topics that are not dealt with sufficiently thoroughly from the standpoint of this Handbook in the available literature. … ► ►►►►

$(a,b]$ or $[a,b)$	half-closed intervals.
…
$\mathbb{N}$	set of all positive integers.
…
$\mathbb{Q}$	set of all rational numbers.
…

… ►For equations or other technical information that appeared previously in AMS 55, the DLMF usually includes the corresponding AMS 55 equation number, or other form of reference, together with corrections, if needed. …

12: 26.11 Integer Partitions: Compositions

… ►

c\left(n\right)

denotes the number of compositions of

n

, and

c_{m}\left(n\right)

is the number of compositions into exactly

m

parts.

c\left(\in\!T,n\right)

is the number of compositions of

n

with no 1’s, where again

T=\{2,3,4,\ldots\}

. … ►

26.11.1

c\left(0\right)=c\left(\in\!T,0\right)=1.

ⓘ

Symbols:: $\in$ : element of, $c\left(\NVar{n}\right)$ : number of compositions of $n$ , $c\left(\NVar{\mathrm{condition}},\NVar{n}\right)$ : restricted number of compositions of $n$ and $T$ : set
Permalink:: http://dlmf.nist.gov/26.11.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §26.11 and Ch.26

… ►The Fibonacci numbers are determined recursively by … ►Additional information on Fibonacci numbers can be found in Rosen et al. (2000, pp. 140–145).

13: 27.18 Methods of Computation: Primes

§27.18 Methods of Computation: Primes

►An overview of methods for precise counting of the number of primes not exceeding an arbitrary integer

x

is given in Crandall and Pomerance (2005, §3.7). …An analytic approach using a contour integral of the Riemann zeta function (§25.2(i)) is discussed in Borwein et al. (2000). … ►These algorithms are used for testing primality of Mersenne numbers,

2^{n}-1

, and Fermat numbers,

2^{2^{n}}+1

. …

14: 26.6 Other Lattice Path Numbers

§26.6 Other Lattice Path Numbers

… ►

Delannoy Number $D(m,n)$

… ►

Motzkin Number $M(n)$

… ►

Narayana Number $N(n,k)$

… ►

§26.6(iv) Identities

…

15: 24.15 Related Sequences of Numbers

§24.15 Related Sequences of Numbers

►

§24.15(i) Genocchi Numbers

… ►

§24.15(ii) Tangent Numbers

… ►

§24.15(iii) Stirling Numbers

… ►

§24.15(iv) Fibonacci and Lucas Numbers

…

16: 26.5 Lattice Paths: Catalan Numbers

§26.5 Lattice Paths: Catalan Numbers

►

§26.5(i) Definitions

►

C\left(n\right)

is the Catalan number. … ►

§26.5(ii) Generating Function

… ►

§26.5(iii) Recurrence Relations

…

17: 4.19 Maclaurin Series and Laurent Series

… ►In (4.19.3)–(4.19.9),

B_{n}

are the Bernoulli numbers and

E_{n}

are the Euler numbers (§§24.2(i)–24.2(ii)). ►

4.19.3

\tan z=z+\frac{z^{3}}{3}+\frac{2}{15}z^{5}+\frac{17}{315}z^{7}+\cdots+\frac{(-% 1)^{n-1}2^{2n}(2^{2n}-1)B_{2n}}{(2n)!}z^{2n-1}+\cdots,

|z|<\frac{1}{2}\pi

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $\tan\NVar{z}$ : tangent function, $n$ : integer and $z$ : complex variable
A&S Ref:: 4.3.67
Referenced by:: §4.19
Permalink:: http://dlmf.nist.gov/4.19.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §4.19 and Ch.4

►

4.19.4

\csc z=\frac{1}{z}+\frac{z}{6}+\frac{7}{360}z^{3}+\frac{31}{15120}z^{5}+\cdots% +\frac{(-1)^{n-1}2(2^{2n-1}-1)B_{2n}}{(2n)!}z^{2n-1}+\cdots,

0<|z|<\pi

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $\csc\NVar{z}$ : cosecant function, $!$ : factorial (as in $n!$ ), $n$ : integer and $z$ : complex variable
A&S Ref:: 4.3.68
Referenced by:: §4.33
Permalink:: http://dlmf.nist.gov/4.19.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §4.19 and Ch.4

►

4.19.5

\sec z=1+\frac{z^{2}}{2}+\frac{5}{24}z^{4}+\frac{61}{720}z^{6}+\cdots+\frac{(-% 1)^{n}E_{2n}}{(2n)!}z^{2n}+\cdots,

|z|<\frac{1}{2}\pi

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $\sec\NVar{z}$ : secant function, $n$ : integer and $z$ : complex variable
A&S Ref:: 4.3.69
Referenced by:: §24.1, §24.19(i), §24.2(ii)
Permalink:: http://dlmf.nist.gov/4.19.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §4.19 and Ch.4

►

4.19.6

\cot z=\frac{1}{z}-\frac{z}{3}-\frac{z^{3}}{45}-\frac{2}{945}z^{5}-\cdots-% \frac{(-1)^{n-1}2^{2n}B_{2n}}{(2n)!}z^{2n-1}-\cdots,

0<|z|<\pi

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cot\NVar{z}$ : 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

…

18: 26.14 Permutations: Order Notation

… ►As an example,

35247816

is an element of

\mathfrak{S}_{8}.

The inversion number is the number of pairs of elements for which the larger element precedes the smaller: … ► ►The Eulerian number, denoted

\genfrac{<}{>}{0.0pt}{}{n}{k}

, is the number of permutations in

\mathfrak{S}_{n}

with exactly

k

descents. …The Eulerian number

\genfrac{<}{>}{0.0pt}{}{n}{k}

is equal to the number of permutations in

\mathfrak{S}_{n}

with exactly

k

excedances. … ►

§26.14(iii) Identities

…

19: 26.7 Set Partitions: Bell Numbers

§26.7 Set Partitions: Bell Numbers

►

§26.7(i) Definitions

… ►

§26.7(ii) Generating Function

… ►

§26.7(iii) Recurrence Relation

… ►

§26.7(iv) Asymptotic Approximation

…

20: 26.8 Set Partitions: Stirling Numbers

§26.8 Set Partitions: Stirling Numbers

►

§26.8(i) Definitions

… ► … ►

§26.8(v) Identities

… ►

§26.8(vi) Relations to Bernoulli Numbers

…