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21: Charles W. Clark

… ►Civil Service, Archie Mahan Prize of the OSA, the Physical Sciences Award of the Washington Academy of Sciences, the Gold and Silver Medals of the U. …

22: 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

…

23: 24.19 Methods of Computation

… ►

§24.19(i) Bernoulli and Euler Numbers and Polynomials

►Equations (24.5.3) and (24.5.4) enable

B_{n}

and

E_{n}

to be computed by recurrence. …A similar method can be used for the Euler numbers based on (4.19.5). … ►

§24.19(ii) Values of $B_{n}$ Modulo $p$

… ►We list here three methods, arranged in increasing order of efficiency. …

►Reed et al. (1990, pp. 458–470) describes a number-theoretic approach to Fourier analysis (called the arithmetic Fourier transform) that uses the Möbius inversion (27.5.7) to increase efficiency in computing coefficients of Fourier series. ►Congruences are used in constructing perpetual calendars, splicing telephone cables, scheduling round-robin tournaments, devising systematic methods for storing computer files, and generating pseudorandom numbers. … ►There are also applications of number theory in many diverse areas, including physics, biology, chemistry, communications, and art. …

25: 24.10 Arithmetic Properties

§24.10 Arithmetic Properties

… ►Here and elsewhere two rational numbers are congruent if the modulus divides the numerator of their difference. ►

§24.10(ii) Kummer Congruences

… ►

§24.10(iii) Voronoi’s Congruence

… ►

§24.10(iv) Factors

…

26: 26.21 Tables

§26.21 Tables

►Abramowitz and Stegun (1964, Chapter 24) tabulates binomial coefficients

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

for

m

up to 50 and

n

up to 25; extends Table 26.4.1 to

n=10

; tabulates Stirling numbers of the first and second kinds,

s\left(n,k\right)

and

S\left(n,k\right)

, for

n

up to 25 and

k

up to

n

; tabulates partitions

p\left(n\right)

and partitions into distinct parts

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

for

n

up to 500. ►Andrews (1976) contains tables of the number of unrestricted partitions, partitions into odd parts, partitions into parts

\not\equiv\pm 2\pmod{5}

, partitions into parts

\not\equiv\pm 1\pmod{5}

, and unrestricted plane partitions up to 100. It also contains a table of Gaussian polynomials up to

\genfrac{[}{]}{0.0pt}{}{12}{6}_{q}

. ►Goldberg et al. (1976) contains tables of binomial coefficients to

n=100

and Stirling numbers to

n=40

27: Bibliography S

… ►

M. R. Schroeder (2006) Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity. 4th edition, Springer-Verlag, Berlin.

ⓘ

Notes:: Volume 7 in Springer Series in Information Sciences.
External Links:: ZentralBlatt
Referenced by:: §27.16, §27.17.
Encodings:: BibTeX

… ►

M. E. Sherry (1959) The zeros and maxima of the Airy function and its first derivative to 25 significant figures. Report AFCRC-TR-59-135, ASTIA Document No. AD214568 Air Research and Development Command, U.S. Air Force, Bedford, MA.

ⓘ

Notes:: Available from U.S. Department of Commerce, Office of Technical Services, Washington, D.C.
Referenced by:: 2nd item, §9.8(iv).
Encodings:: BibTeX

… ►

I. Sh. Slavutskiĭ (1995) Staudt and arithmetical properties of Bernoulli numbers. Historia Sci. (2) 5 (1), pp. 69–74.

ⓘ

External Links:: ISSN 0285-4821, MathReview (Peter M. Neumann), ZentralBlatt
Referenced by:: §24.10(ii).
Encodings:: BibTeX

►

I. Sh. Slavutskiĭ (1999) About von Staudt congruences for Bernoulli numbers. Comment. Math. Univ. St. Paul. 48 (2), pp. 137–144.

ⓘ

External Links:: ISSN 0010-258X, MathReview (Arnold M. Adelberg), ZentralBlatt
Referenced by:: §24.10(ii).
Encodings:: BibTeX

►

I. Sh. Slavutskiĭ (2000) On the generalized Bernoulli numbers that belong to unequal characters. Rev. Mat. Iberoamericana 16 (3), pp. 459–475.

ⓘ

External Links:: ISSN 0213-2230, MathReview (Aichi Kudo), ZentralBlatt
Referenced by:: §24.4(iii).
Encodings:: BibTeX

…

28: 24.14 Sums

§24.14 Sums

►

§24.14(i) Quadratic Recurrence Relations

… ►

24.14.2

\sum_{k=0}^{n}{n\choose k}B_{k}B_{n-k}=(1-n)B_{n}-nB_{n-1}.

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : integer and $n$ : integer
Permalink:: http://dlmf.nist.gov/24.14.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §24.14(i), §24.14 and Ch.24

… ►

§24.14(ii) Higher-Order Recurrence Relations

… ►For other sums involving Bernoulli and Euler numbers and polynomials see Hansen (1975, pp. 331–347) and Prudnikov et al. (1990, pp. 383–386).

29: 26.1 Special Notation

… ► ►►►►►

$\genfrac{(}{)}{0.0pt}{}{m}{n}$	binomial coefficient.
…
$\genfrac{<}{>}{0.0pt}{}{m}{n}$	Eulerian number.
…
$B\left(n\right)$	Bell number.
$C\left(n\right)$	Catalan number.
…

… ►Other notations for

s\left(n,k\right)

, the Stirling numbers of the first kind, include

S_{n}^{(k)}

(Abramowitz and Stegun (1964, Chapter 24), Fort (1948)),

S_{n}^{k}

(Jordan (1939), Moser and Wyman (1958a)),

\genfrac{(}{)}{0.0pt}{}{n-1}{k-1}B_{n-k}^{(n)}

(Milne-Thomson (1933)),

(-1)^{n-k}S_{1}(n-1,n-k)

(Carlitz (1960), Gould (1960)),

(-1)^{n-k}\left[n\atop k\right]

(Knuth (1992), Graham et al. (1994), Rosen et al. (2000)). ►Other notations for

S\left(n,k\right)

, the Stirling numbers of the second kind, include

\mathscr{S}^{(k)}_{n}

(Fort (1948)),

\mathfrak{S}_{n}^{k}

(Jordan (1939)),

\sigma_{n}^{k}

(Moser and Wyman (1958b)),

\genfrac{(}{)}{0.0pt}{}{n}{k}B_{n-k}^{(-k)}

(Milne-Thomson (1933)),

S_{2}(k,n-k)

(Carlitz (1960), Gould (1960)),

\left\{n\atop k\right\}

(Knuth (1992), Graham et al. (1994), Rosen et al. (2000)), and also an unconventional symbol in Abramowitz and Stegun (1964, Chapter 24).

30: 27.3 Multiplicative Properties

§27.3 Multiplicative Properties

►Except for

\nu\left(n\right)

\Lambda\left(n\right)

p_{n}

, and

\pi\left(x\right)

, the functions in §27.2 are multiplicative, which means

f(1)=1

and … ►

27.3.2

f(n)=\prod_{r=1}^{\nu\left(n\right)}f(p^{a_{r}}_{r}).

ⓘ

Symbols:: $\nu\left(\NVar{n}\right)$ : number of distinct primes dividing a number, $n$ : positive integer, $p,p_{1},\ldots$ : prime numbers, $a_{r}$ : positive exponent and $f(n)$ : multiplicative function
Referenced by:: §27.20, §27.20, §27.3
Permalink:: http://dlmf.nist.gov/27.3.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §27.3 and Ch.27

… ►

27.3.6

\sigma_{\alpha}\left(n\right)=\prod_{r=1}^{\nu\left(n\right)}\frac{p^{\alpha(1% +a_{r})}_{r}-1}{p^{\alpha}_{r}-1},

\alpha\neq 0

ⓘ

Symbols:: $\sigma_{\NVar{\alpha}}\left(\NVar{n}\right)$ : sum of powers of divisors of a number, $\nu\left(\NVar{n}\right)$ : number of distinct primes dividing a number, $n$ : positive integer, $p,p_{1},\ldots$ : prime numbers and $a_{r}$ : positive exponent
A&S Ref:: 24.3.3 I.C
Permalink:: http://dlmf.nist.gov/27.3.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §27.3 and Ch.27

… ►

27.3.10

f(n)=\prod_{r=1}^{\nu\left(n\right)}\left(f(p_{r})\right)^{a_{r}}.

ⓘ

Symbols:: $\nu\left(\NVar{n}\right)$ : number of distinct primes dividing a number, $n$ : positive integer, $p,p_{1},\ldots$ : prime numbers, $a_{r}$ : positive exponent and $f(n)$ : multiplicative function
Referenced by:: §27.20, §27.20
Permalink:: http://dlmf.nist.gov/27.3.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §27.3 and Ch.27

”Blockchain” Customer service Number 1206⤄360⤄1604 contact ☎️"Number"

21—30 of 237 matching pages

21: Charles W. Clark

22: 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

23: 24.19 Methods of Computation

§24.19(i) Bernoulli and Euler Numbers and Polynomials

§24.19(ii) Values of $B_{n}$ Modulo $p$

24: 27.17 Other Applications

§27.17 Other Applications

25: 24.10 Arithmetic Properties

§24.10 Arithmetic Properties

§24.10(ii) Kummer Congruences

§24.10(iii) Voronoi’s Congruence

§24.10(iv) Factors

26: 26.21 Tables

§26.21 Tables

27: Bibliography S

28: 24.14 Sums

§24.14 Sums

§24.14(i) Quadratic Recurrence Relations

§24.14(ii) Higher-Order Recurrence Relations

29: 26.1 Special Notation

30: 27.3 Multiplicative Properties

§27.3 Multiplicative Properties

”Blockchain” Customer service Number 1206⤄360⤄1604 contact ☎️"Number"

21—30 of 237 matching pages

21: Charles W. Clark

22: 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

23: 24.19 Methods of Computation

§24.19(i) Bernoulli and Euler Numbers and Polynomials

§24.19(ii) Values of B n Modulo p

24: 27.17 Other Applications

§27.17 Other Applications

25: 24.10 Arithmetic Properties

§24.10 Arithmetic Properties

§24.10(ii) Kummer Congruences

§24.10(iii) Voronoi’s Congruence

§24.10(iv) Factors

26: 26.21 Tables

§26.21 Tables

27: Bibliography S

28: 24.14 Sums

§24.14 Sums

§24.14(i) Quadratic Recurrence Relations

§24.14(ii) Higher-Order Recurrence Relations

29: 26.1 Special Notation

30: 27.3 Multiplicative Properties

§27.3 Multiplicative Properties

§24.19(ii) Values of $B_{n}$ Modulo $p$