finite sum

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1: 34.6 Definition: $\mathit{9j}$ Symbol

… ►The

\mathit{9j}

symbol may be defined either in terms of

\mathit{3j}

symbols or equivalently in terms of

\mathit{6j}

symbols: ►

34.6.1

\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{\mbox{\scriptsize all }m_{rs}}\begin{% pmatrix}j_{11}&j_{12}&j_{13}\\ m_{11}&m_{12}&m_{13}\end{pmatrix}\begin{pmatrix}j_{21}&j_{22}&j_{23}\\ m_{21}&m_{22}&m_{23}\end{pmatrix}\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{31}&m_{32}&m_{33}\end{pmatrix}\*\begin{pmatrix}j_{11}&j_{21}&j_{31}\\ m_{11}&m_{21}&m_{31}\end{pmatrix}\begin{pmatrix}j_{12}&j_{22}&j_{32}\\ m_{12}&m_{22}&m_{32}\end{pmatrix}\begin{pmatrix}j_{13}&j_{23}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix},

ⓘ

Defines:: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol
Symbols:: $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $r$ : nonnegative integer and $s$ : integer
Permalink:: http://dlmf.nist.gov/34.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §34.6 and Ch.34

… ►The

\mathit{9j}

symbol may also be written as a finite triple sum equivalent to a terminating generalized hypergeometric series of three variables with unit arguments. …

2: 15.15 Sums

… ►For compendia of finite sums and infinite series involving hypergeometric functions see Prudnikov et al. (1990, §§5.3 and 6.7) and Hansen (1975). …

3: 25.15 Dirichlet $L$ -functions

… ►

25.15.3

L\left(s,\chi\right)=k^{-s}\sum_{r=1}^{k-1}\chi(r)\zeta\left(s,\frac{r}{k}% \right),

ⓘ

Symbols:: $L\left(\NVar{s},\NVar{\chi}\right)$ : Dirichlet $L$ -function, $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, $k$ : nonnegative integer, $s$ : complex variable and $\chi(n)$ : Dirichlet character
Keywords:: finitesum
Source:: Apostol (1976, (8), p. 255)
Notes:: In the original source the upper-index of the finite sum is $k$ . We use $k-1$ since $\chi(k)=0$ .
Referenced by:: (25.11.36), (25.11.36), §25.11(x), §25.15(i), Erratum (V1.0.23) for Equation (25.11.36)
Permalink:: http://dlmf.nist.gov/25.15.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §25.15(i), §25.15 and Ch.25

… ►

25.15.6

G(\chi)\equiv\sum_{r=1}^{k-1}\chi(r)e^{2\pi ir/k}.

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\equiv$ : equals by definition, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $k$ : nonnegative integer and $\chi(n)$ : Dirichlet character
Keywords:: definition
Source:: Apostol (1976, p. 262)
Referenced by:: Erratum (V1.1.4) for Equation (25.15.6)
Permalink:: http://dlmf.nist.gov/25.15.E6
Encodings:: pMML, png, TeX
Clarification (effective with 1.1.4):: The upper-index of the finitesum which originally was $k$ , was replaced with $k-1$ since $\chi(k)=0$ .
Suggested 2021-08-23 by Gergő Nemes
See also:: Annotations for §25.15(i), §25.15 and Ch.25

… ►

25.15.10

L\left(0,\chi\right)=\begin{cases}\displaystyle-\frac{1}{k}\sum_{r=1}^{k-1}r% \chi(r),&\chi\neq\chi_{1},\\ 0,&\chi=\chi_{1}.\end{cases}

ⓘ

Symbols:: $L\left(\NVar{s},\NVar{\chi}\right)$ : Dirichlet $L$ -function, $k$ : nonnegative integer and $\chi(n)$ : Dirichlet character
Keywords:: zeros
Source:: Apostol (1976, Theorem 12.20, p. 268)
Referenced by:: Erratum (V1.1.4) for Equation (25.15.10)
Permalink:: http://dlmf.nist.gov/25.15.E10
Encodings:: pMML, png, TeX
Clarification (effective with 1.1.4):: The upper-index of the finitesum which originally was $k$ , was replaced with $k-1$ since $\chi(k)=0$ .
Suggested 2021-08-23 by Gergő Nemes
See also:: Annotations for §25.15(ii), §25.15 and Ch.25

4: 34.2 Definition: $\mathit{3j}$ Symbol

… ►When both conditions are satisfied the

\mathit{3j}

symbol can be expressed as the finite sum … ►where

{{}_{3}F_{2}}

is defined as in §16.2. ►For alternative expressions for the

\mathit{3j}

symbol, written either as a finite sum or as other terminating generalized hypergeometric series

{{}_{3}F_{2}}

of unit argument, see Varshalovich et al. (1988, §§8.21, 8.24–8.26).

5: 34.4 Definition: $\mathit{6j}$ Symbol

§34.4 Definition: $\mathit{6j}$ Symbol

… ►The

\mathit{6j}

symbol can be expressed as the finite sum … ►For alternative expressions for the

\mathit{6j}

symbol, written either as a finite sum or as other terminating generalized hypergeometric series

{{}_{4}F_{3}}

of unit argument, see Varshalovich et al. (1988, §§9.2.1, 9.2.3).

6: 25.16 Mathematical Applications

… ►

25.16.6

H\left(s\right)=-\zeta'\left(s\right)+\gamma\zeta\left(s\right)+\frac{1}{2}% \zeta\left(s+1\right)+\sum_{r=1}^{k}\zeta\left(1-2r\right)\zeta\left(s+2r% \right)+\sum_{n=1}^{\infty}\frac{1}{n^{s}}\int_{n}^{\infty}\frac{\widetilde{B}% _{2k+1}\left(x\right)}{x^{2k+2}}\,\mathrm{d}x,

ⓘ

Symbols:: $\gamma$ : Euler’s constant, $H\left(\NVar{s}\right)$ : Euler sums, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)$ : periodic Bernoulli functions, $k$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable and $s$ : complex variable
Keywords:: Euler sum, finitesum, improper integral, infinite series, integral representation, series representation
Source:: Apostol and Vu (1984, (3), p. 87)
Permalink:: http://dlmf.nist.gov/25.16.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §25.16(ii), §25.16 and Ch.25

►

25.16.7

H\left(s\right)=\frac{1}{2}\zeta\left(s+1\right)+\frac{\zeta\left(s\right)}{s-% 1}-\sum_{r=1}^{k}\genfrac{(}{)}{0.0pt}{}{s+2r-2}{2r-1}\zeta\left(1-2r\right)% \zeta\left(s+2r\right)-\genfrac{(}{)}{0.0pt}{}{s+2k}{2k+1}\sum_{n=1}^{\infty}% \frac{1}{n}\int_{n}^{\infty}\frac{\widetilde{B}_{2k+1}\left(x\right)}{x^{s+2k+% 1}}\,\mathrm{d}x.

ⓘ

Symbols:: $H\left(\NVar{s}\right)$ : Euler sums, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)$ : periodic Bernoulli functions, $k$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable and $s$ : complex variable
Keywords:: Euler sum, finitesum, improper integral, infinite series, integral representation, series representation
Source:: Apostol and Vu (1984, (6), p. 88)
Permalink:: http://dlmf.nist.gov/25.16.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §25.16(ii), §25.16 and Ch.25

… ►when both

H\left(s,z\right)

and

H\left(z,s\right)

are finite. …

7: 5.16 Sums

… ►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).

8: 1.7 Inequalities

… ►

§1.7(i) Finite Sums

…

9: 27.10 Periodic Number-Theoretic Functions

… ►Every function periodic (mod

k

) can be expressed as a finite Fourier series of the form …An example is Ramanujan’s sum: … ►is a periodic function of

n\pmod{k}

and has the finite Fourier-series expansion … ►

G\left(n,\chi\right)

is separable for some

n

if … ►The finite Fourier expansion of a primitive Dirichlet character

\chi\pmod{k}

has the form …

10: Bibliography R

… ►

K. Rottbrand (2000) Finite-sum rules for Macdonald’s functions and Hankel’s symbols. Integral Transform. Spec. Funct. 10 (2), pp. 115–124.

ⓘ

External Links:: ISSN 1065-2469, Document, MathReview, ZentralBlatt
Referenced by:: §10.60(iii).
Encodings:: BibTeX

…

finite sum

1—10 of 75 matching pages

1: 34.6 Definition: 9 ⁢ j Symbol

2: 15.15 Sums

3: 25.15 Dirichlet L -functions

4: 34.2 Definition: 3 ⁢ j Symbol

5: 34.4 Definition: 6 ⁢ j Symbol

§34.4 Definition: 6 ⁢ j Symbol

6: 25.16 Mathematical Applications

7: 5.16 Sums

8: 1.7 Inequalities

§1.7(i) Finite Sums

9: 27.10 Periodic Number-Theoretic Functions

10: Bibliography R

1: 34.6 Definition: $\mathit{9j}$ Symbol

3: 25.15 Dirichlet $L$ -functions

4: 34.2 Definition: $\mathit{3j}$ Symbol

5: 34.4 Definition: $\mathit{6j}$ Symbol

§34.4 Definition: $\mathit{6j}$ Symbol