Dedekind%20sums

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11: 24.20 Tables

§24.20 Tables

►Abramowitz and Stegun (1964, Chapter 23) includes exact values of ${\sum }_{k=1}^m{k}^n$, $m=1(1)100$, $n=1(1)10$; ${\sum }_{k=1}^{\mathrm{\infty }}{k}^{-n}$, ${\sum }_{k=1}^{\mathrm{\infty }}{(-1)}^{k-1}{k}^{-n}$, ${\sum }_{k=0}^{\mathrm{\infty }}{(2k+1)}^{-n}$, $n=1,2,\mathrm{\dots }$, 20D; ${\sum }_{k=0}^{\mathrm{\infty }}{(-1)}^k{(2k+1)}^{-n}$, $n=1,2,\mathrm{\dots }$, 18D. ►Wagstaff (1978) gives complete prime factorizations of $N_n$ and $E_n$ for $n=20(2)60$ and $n=8(2)42$, respectively. …

12: Bibliography M

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Affine root systems and Dedekind’s $\eta$-function. Invent. Math. 15 (2), pp. 91–143.

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External Links:

Document, MathReview (D.-N. Verma), ZentralBlatt

Referenced by:

§20.12(i).

Encodings:

BibTeX

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Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.

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Notes:

Includes Fortran code, maximum accuracy 20S.

External Links:

ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§6.13, 4th item, §6.21(ii).

Encodings:

BibTeX

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Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.

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External Links:

ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§10.74(iii).

Encodings:

BibTeX

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Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems. Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.

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External Links:

FullText

Referenced by:

§8.24(i).

Encodings:

BibTeX

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The $q$-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

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External Links:

ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.27(v).

Encodings:

BibTeX

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13: 20 Theta Functions

Chapter 20 Theta Functions

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14: 20.3 Graphics

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Figure 20.3.2: ${\theta }_1(\pi x,q)$, $0\le x\le 2$, $q$ = 0. …For $q\le q^{\text{<span class="ltx\_text hilite">Dedekind</span>}}$, ${\theta }_1(\pi x,q)$ is convex in $x$ for . Here $q^{\text{<span class="ltx\_text hilite">Dedekind</span>}}={\mathrm{e}}^{-\pi y_0}=0.19$ approximately, where $y=y_0$ corresponds to the maximum value of Dedekind’s eta function $\eta \left(\mathrm{i}y\right)$ as depicted in Figure 23.16.1. Magnify

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15: 27.2 Functions

… ►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. …It can be expressed as a sum over all primes $p\le x$: … ►the sum of the $k$th powers of the positive integers $m\le n$ 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. … ►Table 27.2.2 tabulates the Euler totient function $\varphi \left(n\right)$, the divisor function $d\left(n\right)$ ($={\sigma }_0(n)$), and the sum of the divisors $\sigma (n)$ ($={\sigma }_1(n)$), for $n=1(1)52$. …

16: 25.12 Polylogarithms

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25.12.7 $${\mathrm{Li}}_2\left({\mathrm{e}}^{\mathrm{i}\theta }\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{\cos \left(n\theta \right)}{n^2}+\mathrm{i}\sum _{n=1}^{\mathrm{\infty }}\frac{\sin \left(n\theta \right)}{n^2}.$$

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Symbols:

$\cos z$: cosine function, ${\mathrm{Li}}_2\left(z\right)$: dilogarithm, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $n$: nonnegative integer and $\theta$: phase

Keywords:

infinite series, series representation

Source:

Maximon (2003, (8.7), p. 2813)

Referenced by:

§25.12(i)

Permalink:

http://dlmf.nist.gov/25.12.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §25.12(i), §25.12 and Ch.25

►The cosine series in (25.12.7) has the elementary sum … ►

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Figure 25.12.1: Dilogarithm function ${\mathrm{Li}}_2\left(x\right)$, Magnify

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Figure 25.12.2: Absolute value of the dilogarithm function $\left|{\mathrm{Li}}_2\left(x+\mathrm{i}y\right)\right|$, $-20\le x\le 20$, $-20\le y\le 20$. … Magnify 3D Help

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25.12.10 $${\mathrm{Li}}_s\left(z\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{z^n}{n^s}.$$

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Defines:

${\mathrm{Li}}_s\left(z\right)$: polylogarithm

Symbols:

$n$: nonnegative integer, $s$: complex variable and $z$: complex variable

Keywords:

definition, infinite series, series representation

Source:

Lewin (1981, p. 236)

Referenced by:

(25.14.3)

Permalink:

http://dlmf.nist.gov/25.12.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §25.12(ii), §25.12 and Ch.25

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17: 4.27 Sums

§4.27 Sums

►For sums of trigonometric and inverse trigonometric functions see Gradshteyn and Ryzhik (2000, Chapter 1), Hansen (1975, §§14–42), Oberhettinger (1973), and Prudnikov et al. (1986a, Chapter 5).

18: 6.16 Mathematical Applications

… ►The $n$th partial sum is given by ►

6.16.2 $$S_n(x)=\sum _{k=0}^{n-1}\frac{\sin \left((2k+1)x\right)}{2k+1}=\frac{1}{2}{\int }_0^x\frac{\sin \left(2nt\right)}{\sin t}dt=\frac{1}{2}\mathrm{Si}\left(2nx\right)+R_n(x),$$

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Defines:

$S_n(x)$: partial sum (locally)

Symbols:

$dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $\mathrm{Si}\left(z\right)$: sine integral, $x$: real variable, $n$: nonnegative integer and $R_n(x)$: remainder term

Permalink:

http://dlmf.nist.gov/6.16.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.16(i), §6.16 and Ch.6

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Figure 6.16.2: The logarithmic integral $\mathrm{li}\left(x\right)$, together with vertical bars indicating the value of $\pi (x)$ for $x=10,20,\mathrm{\dots },1000$. Magnify

19: 26.3 Lattice Paths: Binomial Coefficients

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26.3.3 $$\sum _{n=0}^m\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)x^n={(1+x)}^m,$$ $m=0,1,\mathrm{\dots }$,

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $x$: real variable, $m$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

24.1.1

Permalink:

http://dlmf.nist.gov/26.3.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §26.3(ii), §26.3 and Ch.26

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26.3.4 $$\sum _{m=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{m+n}{m}\right)x^m=\frac{1}{{(1-x)}^{n+1}},$$ .

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $x$: real variable, $m$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

24.1.1 (in slightly different form)

Permalink:

http://dlmf.nist.gov/26.3.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §26.3(ii), §26.3 and Ch.26

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26.3.7 $$\left(\genfrac{}{}{0.0pt}{}{m+1}{n+1}\right)=\sum _{k=n}^m\left(\genfrac{}{}{0.0pt}{}{k}{n}\right),$$ $m\ge n\ge 0$,

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Symbols:

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

Permalink:

http://dlmf.nist.gov/26.3.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §26.3(iii), §26.3 and Ch.26

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26.3.8 $$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)=\sum _{k=0}^n\left(\genfrac{}{}{0.0pt}{}{m-n-1+k}{k}\right),$$ $m\ge n\ge 0$.

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Symbols:

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

A&S Ref:

24.1.1

Permalink:

http://dlmf.nist.gov/26.3.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §26.3(iii), §26.3 and Ch.26

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26.3.10 $$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)=\sum _{k=0}^n{(-1)}^{n-k}\left(\genfrac{}{}{0.0pt}{}{m+1}{k}\right),$$ $m\ge n\ge 0$,

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Symbols:

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

Permalink:

http://dlmf.nist.gov/26.3.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §26.3(iv), §26.3 and Ch.26

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20: 4.11 Sums

§4.11 Sums

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