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21: 24.11 Asymptotic Approximations

§24.11 Asymptotic Approximations

… ►

24.11.1

(-1)^{n+1}B_{2n}\sim\frac{2(2n)!}{(2\pi)^{2n}},

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ) and $n$ : integer
Referenced by:: §24.11
Permalink:: http://dlmf.nist.gov/24.11.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §24.11 and Ch.24

►

24.11.2

(-1)^{n+1}B_{2n}\sim 4\sqrt{\pi n}\left(\frac{n}{\pi e}\right)^{2n},

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm and $n$ : integer
Permalink:: http://dlmf.nist.gov/24.11.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §24.11 and Ch.24

►

24.11.3

(-1)^{n}E_{2n}\sim\frac{2^{2n+2}(2n)!}{\pi^{2n+1}},

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ) and $n$ : integer
Permalink:: http://dlmf.nist.gov/24.11.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §24.11 and Ch.24

►

24.11.4

(-1)^{n}E_{2n}\sim 8\sqrt{\frac{n}{\pi}}\left(\frac{4n}{\pi e}\right)^{2n}.

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm and $n$ : integer
Referenced by:: §24.11
Permalink:: http://dlmf.nist.gov/24.11.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §24.11 and Ch.24

…

22: 26.9 Integer Partitions: Restricted Number and Part Size

§26.9 Integer Partitions: Restricted Number and Part Size

… ►

p_{k}\left(n\right)

denotes the number of partitions of

n

into at most

k

parts. See Table 26.9.1. … ►It follows that

p_{k}\left(n\right)

also equals the number of partitions of

n

into parts that are less than or equal to

k

. …

24: 26.16 Multiset Permutations

… ►Let

S=\{1^{a_{1}},2^{a_{2}},\ldots,n^{a_{n}}\}

be the multiset that has

a_{j}

copies of

j

1\leq j\leq n

. …The number of elements in

\mathfrak{S}_{S}

is the multinomial coefficient (§26.4)

\genfrac{(}{)}{0.0pt}{}{a_{1}+a_{2}+\cdots+a_{n}}{a_{1},a_{2},\ldots,a_{n}}

. … ►The definitions of inversion number and major index can be extended to permutations of a multiset such as

351322453154\in\mathfrak{S}_{\{1^{2},2^{2},3^{3},4^{2},5^{3}\}}

. Thus

\mathop{\mathrm{inv}}(351322453154)=4+8+0+3+1+1+2+3+1+0+1=24

, and

\mathop{\mathrm{maj}}(351322453154)=2+4+8+9+11=34.

… ►

26.16.1

\genfrac{[}{]}{0.0pt}{}{a_{1}+a_{2}+\cdots+a_{n}}{a_{1},a_{2},\ldots,a_{n}}_{q% }=\prod_{k=1}^{n-1}\genfrac{[}{]}{0.0pt}{}{a_{k}+a_{k+1}+\cdots+a_{n}}{a_{k}}_% {q},

ⓘ

Symbols:: $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -binomial coefficient (or Gaussian polynomial), $\genfrac{[}{]}{0.0pt}{}{\NVar{a_{1}}+\NVar{a_{2}}+\dots+\NVar{a_{n}}}{\NVar{a_% {1}},\NVar{a_{2}},\ldots,\NVar{a_{n}}}_{\NVar{q}}$ : $q$ -multinomial coefficient, $k$ : nonnegative integer, $n$ : nonnegative integer, $a_{j}$ : numbers and $q$ : parameter
Permalink:: http://dlmf.nist.gov/26.16.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §26.16 and Ch.26

…

25: 26.3 Lattice Paths: Binomial Coefficients

… ►

\genfrac{(}{)}{0.0pt}{}{m}{n}

is the number of ways of choosing

n

objects from a collection of

m

distinct objects without regard to order.

\genfrac{(}{)}{0.0pt}{}{m+n}{n}

is the number of lattice paths from

(0,0)

(m,n)

. …The number of lattice paths from

(0,0)

(m,n)

m\leq n

, that stay on or above the line

y=x

\genfrac{(}{)}{0.0pt}{}{m+n}{m}-\genfrac{(}{)}{0.0pt}{}{m+n}{m-1}.

… ►

Table 26.3.1: Binomial coefficients

\genfrac{(}{)}{0.0pt}{}{m}{n}

► ►►►

$m$	$n$
$m$	…
8	1	8	28	56	70	56	28	8	1
…

► ►

Table 26.3.2: Binomial coefficients

\genfrac{(}{)}{0.0pt}{}{m+n}{m}

for lattice paths.

► ►►►

$m$	$n$
$m$	0	1	2	3	4	5	6	7	8
…

► …

26: 8 Incomplete Gamma and Related
Functions

Chapter 8 Incomplete Gamma and Related Functions

…

27: 26.10 Integer Partitions: Other Restrictions

… ►

p\left(\mathcal{D},n\right)

denotes the number of partitions of

n

into distinct parts.

p_{m}\left(\mathcal{D},n\right)

denotes the number of partitions of

n

into at most

m

distinct parts.

p\left(\mathcal{D}k,n\right)

denotes the number of partitions of

n

into parts with difference at least

k

. …

p\left(\mathcal{O},n\right)

denotes the number of partitions of

n

into odd parts.

p\left(\in\!S,n\right)

denotes the number of partitions of

n

into parts taken from the set

S

. …

28: 13.27 Mathematical Applications

… ►where

\alpha

\beta

\gamma

\delta

are real numbers, and

\gamma>0

. Vilenkin (1968, Chapter 8) constructs irreducible representations of this group, in which the diagonal matrices correspond to operators of multiplication by an exponential function. …

… ► 2016) was Professor Emeritus in the Department of Mathematics at the California Institute of Technology (Caltech), Pasadena, California, until his death on May 8, 2016. … ►He was internationally known for his textbooks on calculus, analysis, and analytic number theory, which have been translated into five languages, and for creating Project MATHEMATICS!, a series of video programs that bring mathematics to life with computer animation, live action, music, and special effects. … ►In 1998, the Mathematical Association of America (MAA) awarded him the annual Trevor Evans Award, presented to authors of an exceptional article that is accessible to undergraduates, for his piece entitled “What Is the Most Surprising Result in Mathematics?” (Answer: the prime number theorem). … ►

27 Functions of Number Theory

…

30: 27.10 Periodic Number-Theoretic Functions

§27.10 Periodic Number-Theoretic Functions

►If

k

is a fixed positive integer, then a number-theoretic function

f

is periodic (mod

k

) if … ►is a periodic function of

n\pmod{k}

and has the finite Fourier-series expansion … ►Another generalization of Ramanujan’s sum is the Gauss sum

G\left(n,\chi\right)

associated with a Dirichlet character

\chi\pmod{k}

. … ►

G\left(n,\chi\right)

is separable for some

n

if …

Japan Airlines Reservation Number 📞8

21—30 of 354 matching pages

21: 24.11 Asymptotic Approximations

§24.11 Asymptotic Approximations

22: 26.9 Integer Partitions: Restricted Number and Part Size

§26.9 Integer Partitions: Restricted Number and Part Size

23: 24.16 Generalizations

§24.16 Generalizations

Polynomials and Numbers of Integer Order

Bernoulli Numbers of the Second Kind

Degenerate Bernoulli Numbers

§24.16(iii) Other Generalizations

24: 26.16 Multiset Permutations

25: 26.3 Lattice Paths: Binomial Coefficients

26: 8 Incomplete Gamma and Related
Functions

Chapter 8 Incomplete Gamma and Related Functions

27: 26.10 Integer Partitions: Other Restrictions

28: 13.27 Mathematical Applications

29: Tom M. Apostol

30: 27.10 Periodic Number-Theoretic Functions

§27.10 Periodic Number-Theoretic Functions

Japan Airlines Reservation Number 📞8

21—30 of 354 matching pages

21: 24.11 Asymptotic Approximations

§24.11 Asymptotic Approximations

22: 26.9 Integer Partitions: Restricted Number and Part Size

§26.9 Integer Partitions: Restricted Number and Part Size

23: 24.16 Generalizations

§24.16 Generalizations

Polynomials and Numbers of Integer Order

Bernoulli Numbers of the Second Kind

Degenerate Bernoulli Numbers

§24.16(iii) Other Generalizations

24: 26.16 Multiset Permutations

25: 26.3 Lattice Paths: Binomial Coefficients

26: 8 Incomplete Gamma and Related Functions

Chapter 8 Incomplete Gamma and Related Functions

27: 26.10 Integer Partitions: Other Restrictions

28: 13.27 Mathematical Applications

29: Tom M. Apostol

30: 27.10 Periodic Number-Theoretic Functions

§27.10 Periodic Number-Theoretic Functions

26: 8 Incomplete Gamma and Related
Functions