DLMF: Untitled Document

§24.20 Tables

►Abramowitz and Stegun (1964, Chapter 23) includes exact values of

\sum_{k=1}^{m}k^{n}

,

m=1(1)100

,

n=1(1)10

;

\sum_{k=1}^{\infty}k^{-n}

,

\sum_{k=1}^{\infty}(-1)^{k-1}k^{-n}

,

\sum_{k=0}^{\infty}(2k+1)^{-n}

,

n=1,2,\dotsc

, 20D;

\sum_{k=0}^{\infty}(-1)^{k}(2k+1)^{-n}

,

n=1,2,\dotsc

, 18D. ►Wagstaff (1978) gives complete prime factorizations of

N_{n}

and

E_{n}

for

n=20(2)60

and

n=8(2)42

, respectively. …

… ►Functions in this section derive their properties from the fundamental theorem of arithmetic, which states that every integer

n>1

can be represented uniquely as a product of prime powers, …Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. … ►the sum of the

k

th powers of the positive integers

m\leq n

that are relatively prime to

n

. … ►is the sum of the

\alpha

th powers of the divisors of

n

, where the exponent

\alpha

can be real or complex. … ►where

p^{a}

is a prime power with

a\geq 1

; otherwise

\Lambda\left(n\right)=0

. …

§23.9 Laurent and Other Power Series

… ►

23.9.1

c_{n}=(2n-1)\sum_{w\in\mathbb{L}\setminus\{0\}}w^{-2n},

n=2,3,4,\dots

.

ⓘ

Symbols:: $\in$ : element of, $\setminus$ : set subtraction, $\mathbb{L}$ : lattice, $n$ : integer and $c_{n}$
Referenced by:: §20.6
Permalink:: http://dlmf.nist.gov/23.9.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §23.9 and Ch.23

… ►

23.9.2

\wp\left(z\right)=\frac{1}{z^{2}}+\sum_{n=2}^{\infty}c_{n}z^{2n-2},

0<|z|<|z_{0}|

,

ⓘ

Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $\mathbb{L}$ : lattice, $n$ : integer, $z$ : complex and $c_{n}$
A&S Ref:: 18.5.1
Referenced by:: §23.9, §23.9
Permalink:: http://dlmf.nist.gov/23.9.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §23.9 and Ch.23

… ►

c_{2}=\frac{1}{20}g_{2},

… ►

23.9.5

c_{n}=\frac{3}{(2n+1)(n-3)}\sum_{m=2}^{n-2}c_{m}c_{n-m},

n\geq 4

.

ⓘ

Symbols:: $n$ : integer, $m$ : integer and $c_{n}$
A&S Ref:: 18.5.3
Referenced by:: §23.9
Permalink:: http://dlmf.nist.gov/23.9.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §23.9 and Ch.23

…

… ►

32.8.3

w(z;3)=\frac{3z^{2}}{z^{3}+4}-\frac{6z^{2}(z^{3}+10)}{z^{6}+20z^{3}-80},

ⓘ

Symbols:: $z$ : real
Permalink:: http://dlmf.nist.gov/32.8.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §32.8(ii), §32.8 and Ch.32

►

32.8.4

w(z;4)=-\frac{1}{z}+\frac{6z^{2}(z^{3}+10)}{z^{6}+20z^{3}-80}-\frac{9z^{5}(z^{% 3}+40)}{z^{9}+60z^{6}+11200}.

ⓘ

Symbols:: $z$ : real
Permalink:: http://dlmf.nist.gov/32.8.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §32.8(ii), §32.8 and Ch.32

… ►where the

Q_{n}(z)

are monic polynomials (coefficient of highest power of

z

is

1

) satisfying … ►

Q_{3}(z)=z^{6}+20z^{3}-80,

… ►

32.8.8

\sum_{m=0}^{\infty}p_{m}(z)\lambda^{m}=\exp\left(z\lambda-\tfrac{4}{3}\lambda^% {3}\right).

ⓘ

Symbols:: $\exp\NVar{z}$ : exponential function, $m$ : integer, $z$ : real and $p_{m}(z)$ : polynomials
Permalink:: http://dlmf.nist.gov/32.8.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §32.8(ii), §32.8 and Ch.32

…

… ►

§20.11(i) Gauss Sum

►For relatively prime integers

m, n

with

n>0

and

m n

even, the Gauss sum

G(m,n)

is defined by ►

20.11.1

G(m,n)=\sum\limits_{k=0}^{n-1}e^{-\pi ik^{2}m/n};

ⓘ

Defines:: $G(m,n)$ : Gauss sum (locally)
Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $m$ : integer and $n$ : integer
Permalink:: http://dlmf.nist.gov/20.11.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §20.11(i), §20.11 and Ch.20

… ► … ►In the case

z=0

identities for theta functions become identities in the complex variable

q

, with

\left|q\right|<1

, that involve rational functions, power series, and continued fractions; see Adiga et al. (1985), McKean and Moll (1999, pp. 156–158), and Andrews et al. (1988, §10.7). …

… ►If we permit the use of nonelementary functions as approximants, then even more powerful re-expansions become available. … ►These answers are linked to the terms involving the complementary error function in the more powerful expansions typified by the combination of (2.11.10) and (2.11.15). … ►The first of these two references also provides an introduction to the powerful Borel transform theory. … ►The following example, based on Weniger (1996), illustrates their power. … ►For example, using double precision

d_{20}

is found to agree with (2.11.31) to 13D. …

… ►When the differences are moderately small, the iteration is stopped, the elementary symmetric functions of certain differences are calculated, and a polynomial consisting of a fixed number of terms of the sum in (19.19.7) is evaluated. … ►

19.36.2

1-\tfrac{3}{14}E_{2}+\tfrac{1}{6}E_{3}+\tfrac{9}{88}E_{2}^{2}-\tfrac{3}{22}E_{% 4}-\tfrac{9}{52}E_{2}E_{3}+\tfrac{3}{26}E_{5}-\tfrac{1}{16}E_{2}^{3}+\tfrac{3}% {40}E_{3}^{2}+\tfrac{3}{20}E_{2}E_{4}+\tfrac{45}{272}E_{2}^{2}E_{3}-\tfrac{9}{% 68}(E_{3}E_{4}+E_{2}E_{5}).

ⓘ

Symbols:: $E_{s}(\mathbf{z})$ : symmetric function
Referenced by:: §19.19
Permalink:: http://dlmf.nist.gov/19.36.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §19.36(i), §19.36 and Ch.19

… ►

19.36.13

2R_{G}\left(t_{0}^{2},t_{0}^{2}+\theta c_{0}^{2},t_{0}^{2}+\theta a_{0}^{2}% \right)=\left(t_{0}^{2}+\theta\sum_{m=0}^{\infty}2^{m-1}c_{m}^{2}\right)R_{C}% \left(T^{2}+\theta M^{2},T^{2}\right)+h_{0}+\sum_{m=1}^{\infty}2^{m}(h_{m}-h_{% m-1}).

ⓘ

Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind, $m$ : nonnegative integer, $M$ : limit, $T$ : limit and $h_{n}$
Permalink:: http://dlmf.nist.gov/19.36.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §19.36(ii), §19.36(ii), §19.36 and Ch.19

… ►For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). … ►Faster convergence of power series for

K\left(k\right)

and

E\left(k\right)

can be achieved by using (19.5.1) and (19.5.2) in the right-hand sides of (19.8.12). …

… ►

P. L. Kapitsa (1951b) The computation of the sums of negative even powers of roots of Bessel functions. Doklady Akad. Nauk SSSR (N.S.) 77, pp. 561–564.

ⓘ

External Links:: MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §10.21(xiii).
Encodings:: BibTeX

… ►

R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

ⓘ

External Links:: ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

… ►

M. K. Kerimov (1980) Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

ⓘ

External Links:: Document, MathReview (Matti Jutila), ZentralBlatt
Referenced by:: §25.18(i).
Encodings:: BibTeX

… ►

A. V. Kitaev and A. H. Vartanian (2004) Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

ⓘ

External Links:: ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt
Referenced by:: §32.11(iv).
Encodings:: BibTeX

… ►

C. Krattenthaler (1993) HYP and HYPQ. Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively

q

-binomial sums and basic hypergeometric series. Séminaire Lotharingien de Combinatoire 30, pp. 61–76.

ⓘ

Notes:: Also published in J. Symbolic Comput. (1995), v. 20, no. 5–6, pp. 737–744.
External Links:: Code, Document, MathReview (G. Gasper), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

…

… ►

W. Gautschi (1994) Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.

ⓘ

Notes:: For remark see same journal 24(1998), p. 355.
External Links:: Document, ISSN 0098-3500, GAMS (module 726; package TOMS), ZentralBlatt
Referenced by:: 2nd item, §3.5(v), §3.5(v).
Encodings:: BibTeX

… ►

A. Gil, J. Segura, and N. M. Temme (2014) Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

ⓘ

Notes:: Includes Fortran program claiming 12S accuracy
External Links:: ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt
Referenced by:: 3rd item, §10.77(ix).
Encodings:: BibTeX

… ►

Ya. I. Granovskiĭ, I. M. Lutzenko, and A. S. Zhedanov (1992) Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.

ⓘ

External Links:: ISSN 0003-4916, Link, MathReview (A. O. Barut), ZentralBlatt
Referenced by:: §18.38(iii).
Encodings:: BibTeX

… ►

P. Groeneboom and D. R. Truax (2000) A monotonicity property of the power function of multivariate tests. Indag. Math. (N.S.) 11 (2), pp. 209–218.

ⓘ

External Links:: ISSN 0019-3577, Document, MathReview (Sumitra Purkayastha), ZentralBlatt
Referenced by:: §35.9.
Encodings:: BibTeX

… ►

E. Grosswald (1985) Representations of Integers as Sums of Squares. Springer-Verlag, New York.

ⓘ

External Links:: ISBN 0-387-96126-7, MathReview (Harvey Cohn), ZentralBlatt
Referenced by:: §27.13(iv).
Encodings:: BibTeX

…

sums%20of%20powers

9 matching pages

1: 24.20 Tables

§24.20 Tables

2: 27.2 Functions

3: 23.9 Laurent and Other Power Series

§23.9 Laurent and Other Power Series

4: 32.8 Rational Solutions

5: 20.11 Generalizations and Analogs

§20.11(i) Gauss Sum

6: 2.11 Remainder Terms; Stokes Phenomenon

7: 19.36 Methods of Computation

8: Bibliography K

9: Bibliography G