sums of powers

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11: 24.17 Mathematical Applications

… ►Bernoulli and Euler numbers and polynomials occur in: number theory via (24.4.7), (24.4.8), and other identities involving sums of powers; the Riemann zeta function and $L$-series (§25.15, Apostol (1976), and Ireland and Rosen (1990)); arithmetic of cyclotomic fields and the classical theory of Fermat’s last theorem (Ribenboim (1979) and Washington (1997)); $p$-adic analysis (Koblitz (1984, Chapter 2)). …

12: 27.14 Unrestricted Partitions

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27.14.7 $$np\left(n\right)=\sum _{k=1}^n{\sigma }_1\left(k\right)p\left(n-k\right),$$

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Symbols:

${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $p\left(n\right)$: total number of partitions of $n$ , $k$: positive integer and $n$: positive integer

A&S Ref:

24.2.1 II.A (in slightly different form) 24.2.1 II.A (in slightly different form)

Referenced by:

§27.20, §27.20, Erratum (V1.0.11) for Clarifications

Permalink:

http://dlmf.nist.gov/27.14.E7

Encodings:

pMML, png, TeX

Errata (effective with 1.0.11):

The erroneous ${\sigma }_1\left(n\right)$ in
$np\left(n\right)={\sum }_{k=1}^n{\sigma }_1\left(n\right)p\left(n-k\right)$
was replaced.

Reported 2015-08-17 by Howard Cohl and Hannah Cohen

See also:

Annotations for §27.14(ii), §27.14 and Ch.27

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27.14.20 $$\tau \left(n\right)\equiv {\sigma }_{11}\left(n\right)(mod691).$$

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Symbols:

$\tau \left(n\right)$: Ramanujan’s tau function, ${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $\equiv$: modular equivalence and $n$: positive integer

Referenced by:

§27.20, §27.20

Permalink:

http://dlmf.nist.gov/27.14.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §27.14(vi), §27.14 and Ch.27

…

13: Bibliography H

… ►

Sums of powers of integers via generating functions. Fibonacci Quart. 34 (3), pp. 244–256.

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External Links:

ISSN 0015-0517, MathReview (Noriaki Kimura), ZentralBlatt

Referenced by:

§24.4(iii).

Encodings:

BibTeX

…

14: 3.9 Acceleration of Convergence

… ►If $s_n$ is the $n$th partial sum of a power series $f$, then $t_{n,2k}={\epsilon }_{2k}^{(n)}$ is the Padé approximant ${[(n+k)/k]}_f$ (§3.11(iv)). …

15: 10.65 Power Series

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§10.65(iii) Cross-Products and Sums of Squares

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16: 27.10 Periodic Number-Theoretic Functions

… ►This is the sum of the $n$th powers of the primitive $k$th roots of unity. …

17: Bibliography K

… ►

The computation of the sums of negative even powers of roots of Bessel functions. Doklady Akad. Nauk SSSR (N.S.) 77, pp. 561–564.

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§10.21(xiii).

Encodings:

BibTeX

…

18: 1.9 Calculus of a Complex Variable

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1.9.63 $$f^{(m)}(z)=\sum _{n=0}^{\mathrm{\infty }}{\left(n+1\right)}_m{a}_{n+m}{(z-z_0)}^n,$$ , $m=0,1,2,\mathrm{\dots }$.

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $z$: variable, $m$: nonnegative integer, $n$: nonnegative integer, $R$: radius of convergence and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.9.E63

Encodings:

pMML, png, TeX

See also:

Annotations for §1.9(vi), §1.9(vi), §1.9 and Ch.1

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19: 27.13 Functions

… ►This problem is named after Edward Waring who, in 1770, stated without proof and with limited numerical evidence, that every positive integer $n$ is the sum of four squares, of nine cubes, of nineteen fourth powers, and so on. …

20: 27.21 Tables

… ►Glaisher (1940) contains four tables: Table I tabulates, for all $n\le {10}^4$: (a) the canonical factorization of $n$ into powers of primes; (b) the Euler totient $\varphi \left(n\right)$; (c) the divisor function $d\left(n\right)$; (d) the sum $\sigma (n)$ of these divisors. …