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31: 24.14 Sums

§24.14 Sums

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§24.14(i) Quadratic Recurrence Relations

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24.14.2 $$\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)B_k{B}_{n-k}=(1-n)B_n-n{B}_{n-1}.$$

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Symbols:

$B_n$: Bernoulli numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: 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

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§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).

32: 26.1 Special Notation

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

… ►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).

33: 27.3 Multiplicative Properties

§27.3 Multiplicative Properties

►Except for $\nu \left(n\right)$, $\mathrm{\Lambda }\left(n\right)$, $p_n$, and $\pi \left(x\right)$, the functions in §27.2 are multiplicative, which means $f(1)=1$ and … ►

27.3.2 $$f(n)=\prod _{r=1}^{\nu \left(n\right)}f(p_r^{a_r}).$$

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Symbols:

$\nu \left(n\right)$: number of distinct primes dividing a number, $n$: positive integer, $p,p_1,\mathrm{\dots }$: 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:

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See also:

Annotations for §27.3 and Ch.27

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27.3.6 $${\sigma }_{\alpha }\left(n\right)=\prod _{r=1}^{\nu \left(n\right)}\frac{p_r^{\alpha (1+a_r)}-1}{p_r^{\alpha }-1},$$ $\alpha \ne 0$.

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Symbols:

${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $\nu \left(n\right)$: number of distinct primes dividing a number, $n$: positive integer, $p,p_1,\mathrm{\dots }$: prime numbers and $a_r$: positive exponent

A&S Ref:

24.3.3 I.C

Permalink:

http://dlmf.nist.gov/27.3.E6

Encodings:

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See also:

Annotations for §27.3 and Ch.27

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27.3.10 $$f(n)=\prod _{r=1}^{\nu \left(n\right)}{\left(f(p_r)\right)}^{a_r}.$$

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Symbols:

$\nu \left(n\right)$: number of distinct primes dividing a number, $n$: positive integer, $p,p_1,\mathrm{\dots }$: 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:

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See also:

Annotations for §27.3 and Ch.27

34: 27.1 Special Notation

§27.1 Special Notation

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$d,k,m,n$	positive integers (unless otherwise indicated).
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$p,p_1,p_2,\mathrm{\dots }$	prime numbers (or primes): integers ($>1$) with only two positive integer divisors, $1$ and the number itself.
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$x,y$	real numbers.
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35: Foreword

… ►Particular attention is called to the generous support of the National Science Foundation, which made possible the participation of experts from academia and research institutes worldwide. …

36: 4.48 Software

… ►All scientific programming languages, libraries, and systems support computation of at least some of the elementary functions in standard floating-point arithmetic (§3.1(i)). …

37: 27.12 Asymptotic Formulas: Primes

§27.12 Asymptotic Formulas: Primes

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Prime Number Theorem

… ►The number of such primes not exceeding $x$ is … ►There are infinitely many Carmichael numbers.

38: 27.2 Functions

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§27.2(i) Definitions

… ►where $p_1,p_2,\mathrm{\dots },p_{\nu \left(n\right)}$ are the distinct prime factors of $n$, each exponent $a_r$ is positive, and $\nu \left(n\right)$ is the number of distinct primes dividing $n$. … ►(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) … ► … ►

§27.2(ii) Tables

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39: Bibliography M

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Computational Algebra Group, School of Mathematics and Statistics, University of Sydney, Australia.

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Notes:

A software package (the successor to CAYLEY) designed to solve computationally hard problems in algebra, number theory, geometry, and combinatorics, and to provide a mathematically rigorous environment for computing with algebraic, number-theoretic, combinatoric, and geometric objects.

External Links:

Magma

Referenced by:

4th item.

Encodings:

BibTeX

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The supports of measures associated with orthogonal polynomials and the spectra of the related selfadjoint operators. Rocky Mountain J. Math. 21 (1), pp. 501–527.

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External Links:

ISSN 0035-7596, Document, Link, MathReview (T. S. Chihara)

Referenced by:

§18.2(xi).

Encodings:

BibTeX

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On the representation of numbers as a sum of $2r$ squares. Quarterly Journal of Math. 48, pp. 93–104.

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External Links:

ZentralBlatt

Referenced by:

§27.13(iv).

Encodings:

BibTeX

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Asymptotic development of the Stirling numbers of the first kind. J. London Math. Soc. 33, pp. 133–146.

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External Links:

Document, MathReview (N. J. Fine), ZentralBlatt

Referenced by:

§26.1, §26.8(vii).

Encodings:

BibTeX

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Stirling numbers of the second kind. Duke Math. J. 25 (1), pp. 29–43.

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External Links:

Document, MathReview (A. Sade), ZentralBlatt

Referenced by:

§26.1, §26.8(vii), Notations S.

Encodings:

BibTeX

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40: 24.5 Recurrence Relations

§24.5 Recurrence Relations

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24.5.3 $$\sum _{k=0}^{n-1}\left(\genfrac{}{}{0pt}{}{n}{k}\right)B_k=0,$$ $n=2,3,\mathrm{\dots }$,

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Symbols:

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

Referenced by:

§24.19(i)

Permalink:

http://dlmf.nist.gov/24.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §24.5(i), §24.5 and Ch.24

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24.5.5 $$\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)2^k{E}_{n-k}+E_n=2.$$

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Symbols:

$E_n$: Euler numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer and $n$: integer

Permalink:

http://dlmf.nist.gov/24.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §24.5(i), §24.5 and Ch.24

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§24.5(ii) Other Identities

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§24.5(iii) Inversion Formulas

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