Jordan function

SearchAdvancedHelp

(0.001 seconds)

1—10 of 11 matching pages

1: 27.2 Functions

… ►

27.2.11 $$J_k\left(n\right)=\sum _{\left(\left(d_1,\mathrm{\dots },d_k\right),n\right)=1}1,$$

ⓘ

Defines:

$J_k\left(n\right)$: Jordan’s function

Symbols:

$\left(m,n\right)$: greatest common divisor (gcd), $d$: positive integer, $k$: positive integer and $n$: positive integer

Referenced by:

§27.2(i)

Permalink:

http://dlmf.nist.gov/27.2.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §27.2(i), §27.2 and Ch.27

… ►This is Jordan’s function. …

2: 27.6 Divisor Sums

… ►

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

ⓘ

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:

pMML, png, TeX

See also:

Annotations for §27.6 and Ch.27

►

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

ⓘ

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:

pMML, png, TeX

See also:

Annotations for §27.6 and Ch.27

3: 5.16 Sums

§5.16 Sums

… ►

5.16.2 $$\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k}{\psi }^{\prime }\left(k+1\right)=\zeta \left(3\right)=-\frac{1}{2}{\psi }^{\prime \prime }\left(1\right).$$

ⓘ

Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\psi \left(z\right)$: psi (or digamma) function and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/5.16.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.16 and Ch.5

►For further sums involving the psi function see Hansen (1975, pp. 360–367). For sums of gamma functions see Andrews et al. (1999, Chapters 2 and 3) and §§15.2(i), 16.2. ►For related sums involving finite field analogs of the gamma and beta functions (Gauss and Jacobi sums) see Andrews et al. (1999, Chapter 1) and Terras (1999, pp. 90, 149).

4: 4.18 Inequalities

… ►

Jordan’s Inequality

…

5: 27.3 Multiplicative Properties

… ►

27.3.4 $$J_k\left(n\right)=n^k\prod _{p|n}(1-p^{-k}),$$

ⓘ

Symbols:

$J_k\left(n\right)$: Jordan’s function, $k$: positive integer, $n$: positive integer and $p,p_1,\mathrm{\dots }$: prime numbers

Permalink:

http://dlmf.nist.gov/27.3.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §27.3 and Ch.27

…

6: Peter L. Walker

… ►Walker’s published work has been mainly in real and complex analysis, with excursions into analytic number theory and geometry, the latter in collaboration with Professor Mowaffaq Hajja of the University of Jordan. ►Walker’s books are An Introduction to Complex Analysis, published by Hilger in 1974, The Theory of Fourier Series and Integrals, published by Wiley in 1986, Elliptic Functions. A Constructive Approach, published by Wiley in 1996, and Examples and Theorems in Analysis, published by Springer in 2004. … ►

20 Theta Functions

►

22 Jacobian Elliptic Functions

►

23 Weierstrass Elliptic and Modular Functions

…

7: 24.17 Mathematical Applications

… ►Then with the notation of §24.2(iii) … ►See Milne-Thomson (1933), Nörlund (1924), or Jordan (1965). … ►

§24.17(ii) Spline Functions

… ►The functions … ►

Bernoulli Monosplines

…

8: Bibliography J

… ►

Some properties of the Erlang loss function. Bell System Tech. J. 53, pp. 525–551.

ⓘ

External Links:

ISSN 0005-8580, MathReview (U. Narayan Bhat), ZentralBlatt

Referenced by:

§8.23.

Encodings:

BibTeX

… ►

Asymptotics of the hypergeometric function. Math. Methods Appl. Sci. 24 (6), pp. 369–389.

ⓘ

External Links:

ISSN 0170-4214, Document, MathReview, ZentralBlatt

Referenced by:

§14.15(iii), §15.12(iii).

Encodings:

BibTeX

►

Parabolic cylinder functions of large order. J. Comput. Appl. Math. 190 (1-2), pp. 453–469.

ⓘ

External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§12.10(vi).

Encodings:

BibTeX

… ►

Calculus of Finite Differences. Hungarian Agent Eggenberger Book-Shop, Budapest.

ⓘ

Notes:

Third edition published in 1965 by Chelsea Publishing Co., New York, and American Mathematical Society, AMS Chelsea Book Series, Providence, RI

External Links:

MathReview (W. E. Milne), ZentralBlatt

Referenced by:

§26.1, §26.1, §26.8(vii), §5.16, Notations S.

Encodings:

BibTeX

►

Calculus of Finite Differences. 3rd edition, AMS Chelsea, Providence, RI.

ⓘ

External Links:

MathReview, ZentralBlatt

Referenced by:

§24.17(i), §24.6.

Encodings:

BibTeX

…

9: 26.8 Set Partitions: Stirling Numbers

… ►

§26.8(ii) Generating Functions

… ►

26.8.8 $$\sum _{n=0}^{\mathrm{\infty }}s(n,k)\frac{x^n}{n!}=\frac{{(\ln \left(1+x\right))}^k}{k!},$$ ,

ⓘ

Symbols:

$s(n,k)$: Stirling number of the first kind, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $x$: real variable, $k$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

24.1.3

Permalink:

http://dlmf.nist.gov/26.8.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §26.8(ii), §26.8 and Ch.26

… ►

26.8.13 $$\sum _{n,k=0}^{\mathrm{\infty }}S(n,k)\frac{x^n}{n!}{y}^k=\exp \left(y({\mathrm{e}}^x-1)\right).$$

ⓘ

Symbols:

$S(n,k)$: Stirling number of the second kind, $\exp z$: exponential function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $x$: real variable, $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.8.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §26.8(ii), §26.8 and Ch.26

… ►

26.8.40 $$s(n+1,k+1)\sim {(-1)}^{n-k}\frac{n!}{k!}{(\gamma +\ln n)}^k,$$ $n\to \mathrm{\infty }$,

ⓘ

Symbols:

$\gamma$: Euler’s constant, $s(n,k)$: Stirling number of the first kind, $\sim$: asymptotic equality, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.8.E40

Encodings:

pMML, png, TeX

See also:

Annotations for §26.8(vii), §26.8 and Ch.26

…

10: 26.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) … ►The main functions treated in this chapter are: ► ►►►

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$	binomial coefficient.
…
$p\left(n\right)$	number of partitions of $n$.
…

… ►Other notations for $s(n,k)$, 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)), $\left(\genfrac{}{}{0.0pt}{}{n-1}{k-1}\right)B_{n-k}^{(n)}$ (Milne-Thomson (1933)), ${(-1)}^{n-k}{S}_1(n-1,n-k)$ (Carlitz (1960), Gould (1960)), ${(-1)}^{n-k}\left[\genfrac{}{}{0pt}{}{n}{k}\right]$ (Knuth (1992), Graham et al. (1994), Rosen et al. (2000)). ►Other notations for $S(n,k)$, the Stirling numbers of the second kind, include ${𝒮}_n^{(k)}$ (Fort (1948)), ${𝔖}_n^k$ (Jordan (1939)), ${\sigma }_n^k$ (Moser and Wyman (1958b)), $\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)B_{n-k}^{(-k)}$ (Milne-Thomson (1933)), $S_2(k,n-k)$ (Carlitz (1960), Gould (1960)), $\left\{\genfrac{}{}{0pt}{}{n}{k}\right\}$ (Knuth (1992), Graham et al. (1994), Rosen et al. (2000)), and also an unconventional symbol in Abramowitz and Stegun (1964, Chapter 24).