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

§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

►

34.6.2

\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{j}(-1)^{2j}(2j+1)\begin{Bmatrix}j_{11}% &j_{21}&j_{31}\\ j_{32}&j_{33}&j\end{Bmatrix}\begin{Bmatrix}j_{12}&j_{22}&j_{32}\\ j_{21}&j&j_{23}\end{Bmatrix}\begin{Bmatrix}j_{13}&j_{23}&j_{33}\\ j&j_{11}&j_{12}\end{Bmatrix}.

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol, $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol and $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Referenced by:: §34.13
Permalink:: http://dlmf.nist.gov/34.6.E2
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: 32 Painlevé Transcendents

Chapter 32 Painlevé Transcendents

…

3: 34.7 Basic Properties: $\mathit{9j}$ Symbol

§34.7 Basic Properties: $\mathit{9j}$ Symbol

… ►

§34.7(ii) Symmetry

… ►

§34.7(iv) Orthogonality

… ►

§34.7(vi) Sums

… ►It constitutes an addition theorem for the

\mathit{9j}

symbol. …

4: 34.1 Special Notation

… ► ►►

$2j_{1},2j_{2},2j_{3},2l_{1},2l_{2},2l_{3}$	nonnegative integers.
…

►The main functions treated in this chapter are the Wigner

\mathit{3j},\mathit{6j},\mathit{9j}

symbols, respectively, … ►

\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}.

… ►For other notations for

\mathit{3j}

\mathit{6j}

\mathit{9j}

symbols, see Edmonds (1974, pp. 52, 97, 104–105) and Varshalovich et al. (1988, §§8.11, 9.10, 10.10).

5: 28.15 Expansions for Small $q$

… ►

28.15.1

\lambda_{\nu}\left(q\right)=\nu^{2}+\frac{1}{2(\nu^{2}-1)}q^{2}+\frac{5\nu^{2}% +7}{32(\nu^{2}-1)^{3}(\nu^{2}-4)}q^{4}+\frac{9\nu^{4}+58\nu^{2}+29}{64(\nu^{2}% -1)^{5}(\nu^{2}-4)(\nu^{2}-9)}q^{6}+\cdots.

ⓘ

Symbols:: $\lambda_{\NVar{\nu+2n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $q=h^{2}$ : parameter and $\nu$ : complex parameter
A&S Ref:: 20.3.15 (in slightly different form)
Permalink:: http://dlmf.nist.gov/28.15.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §28.15(i), §28.15 and Ch.28

… ►

28.15.3

\operatorname{me}_{\nu}\left(z,q\right)=e^{\mathrm{i}\nu z}-\frac{q}{4}\left(% \frac{1}{\nu+1}e^{\mathrm{i}(\nu+2)z}-\frac{1}{\nu-1}e^{\mathrm{i}(\nu-2)z}% \right)+\frac{q^{2}}{32}\left(\frac{1}{(\nu+1)(\nu+2)}e^{\mathrm{i}(\nu+4)z}+% \frac{1}{(\nu-1)(\nu-2)}e^{\mathrm{i}(\nu-4)z}-\frac{2(\nu^{2}+1)}{(\nu^{2}-1)% ^{2}}e^{\mathrm{i}\nu z}\right)+\cdots;

ⓘ

Symbols:: $\operatorname{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $q=h^{2}$ : parameter, $z$ : complex variable and $\nu$ : complex parameter
A&S Ref:: 20.3.17 (only two terms and without normalization)
Permalink:: http://dlmf.nist.gov/28.15.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §28.15(ii), §28.15 and Ch.28

…

6: Staff

… ►

Peter A. Clarkson, University of Kent, Chap. 32

… ►

Frank W. J. Olver, University of Maryland and NIST, Chaps. 1, 2, 4, 9, 10

… ►

Peter A. Clarkson, University of Kent, for Chap. 32

… ►

Diego Dominici, State University of New York at New Paltz, for Chaps. 9, 10 (deceased)

…

7: Bibliography E

… ►

M. Edwards, D. A. Griggs, P. L. Holman, C. W. Clark, S. L. Rolston, and W. D. Phillips (1999) Properties of a Raman atom-laser output coupler. J. Phys. B 32 (12), pp. 2935–2950.

ⓘ

External Links:: FullText
Referenced by:: §12.17.
Encodings:: BibTeX

… ►

E. B. Elliott (1903) A formula including Legendre’s

EK^{\prime}+KE^{\prime}-KK^{\prime}=\frac{1}{2}\pi

. Messenger of Math. 33, pp. 31–32.

ⓘ

Notes:: Errata on p. 45 of the same volume.
External Links:: ZentralBlatt
Referenced by:: §15.16.
Encodings:: BibTeX

… ►

A. Erdélyi (1941c) On algebraic Lamé functions. Philos. Mag. (7) 32, pp. 348–350.

ⓘ

External Links:: MathReview (G. Szegő), ZentralBlatt
Referenced by:: §29.17(ii).
Encodings:: BibTeX

… ►

D. Erricolo (2006) Algorithm 861: Fortran 90 subroutines for computing the expansion coefficients of Mathieu functions using Blanch’s algorithm. ACM Trans. Math. Software 32 (4), pp. 622–634.

ⓘ

External Links:: ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: 3rd item, §28.36(iii).
Encodings:: BibTeX

… ►

W. N. Everitt (1982) On the transformation theory of ordinary second-order linear symmetric differential expressions. Czechoslovak Math. J. 32(107) (2), pp. 275–306.

ⓘ

Notes:: With a loose Russian summary
External Links:: ISSN 0011-4642, MathReview (D. Hinton)
Referenced by:: §1.13(viii).
Encodings:: BibTeX

…

8: 26.2 Basic Definitions

… ►

Table 26.2.1: Partitions

p\left(n\right)

► ►►►►

$n$	$p\left(n\right)$	$n$	$p\left(n\right)$	$n$	$p\left(n\right)$
…
9	30	26	2436	43	63261
…
15	176	32	8349	49	1 73525
…

►

9: 28.6 Expansions for Small $q$

… ►Leading terms of the of the power series for

m=7,8,9,\dots

are: ►

28.6.14

\rselection{a_{m}\left(q\right)\\ b_{m}\left(q\right)}=m^{2}+\frac{1}{2(m^{2}-1)}q^{2}+\frac{5m^{2}+7}{32(m^{2}-% 1)^{3}(m^{2}-4)}q^{4}+\frac{9m^{4}+58m^{2}+29}{64(m^{2}-1)^{5}(m^{2}-4)(m^{2}-% 9)}q^{6}+\cdots.

ⓘ

Symbols:: $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $m$ : integer and $q=h^{2}$ : parameter
A&S Ref:: 20.2.26
Referenced by:: §28.6(i), §28.6(i), Erratum (V1.1.10) for Sections 28.6(i), 28.6(ii), 28.8(i), 28.8(ii)
Permalink:: http://dlmf.nist.gov/28.6.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §28.6(i), §28.6 and Ch.28

… ►Numerical values of the radii of convergence

\rho_{n}^{(j)}

of the power series (28.6.1)–(28.6.14) for

n=0,1,\dots,9

are given in Table 28.6.1. … ►

28.6.21

2^{\ifrac{1}{2}}\operatorname{ce}_{0}\left(z,q\right)=1-\tfrac{1}{2}q\cos 2z+% \tfrac{1}{32}q^{2}\left(\cos 4z-2\right)-\tfrac{1}{128}q^{3}\left(\tfrac{1}{9}% \cos 6z-11\cos 2z\right)+\cdots,

ⓘ

Symbols:: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\cos\NVar{z}$ : cosine function, $q=h^{2}$ : parameter and $z$ : complex variable
A&S Ref:: 20.2.27
Referenced by:: §28.6(ii)
Permalink:: http://dlmf.nist.gov/28.6.E21
Encodings:: pMML, png, TeX
See also:: Annotations for §28.6(ii), §28.6 and Ch.28

… ►

28.6.26

\operatorname{ce}_{m}\left(z,q\right)=\cos mz-\frac{q}{4}\left(\frac{1}{m+1}% \cos(m+2)z-\frac{1}{m-1}\cos(m-2)z\right)+\frac{q^{2}}{32}\left(\frac{1}{(m+1)% (m+2)}\cos(m+4)z+\frac{1}{(m-1)(m-2)}\cos(m-4)z-\frac{2(m^{2}+1)}{(m^{2}-1)^{2% }}\cos mz\right)+\cdots.

ⓘ

Symbols:: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\cos\NVar{z}$ : cosine function, $m$ : integer, $q=h^{2}$ : parameter and $z$ : complex variable
A&S Ref:: 20.2.28 (in slightly different form)
Referenced by:: §28.6(ii), §28.6(ii), §28.6(ii), Erratum (V1.1.10) for Sections 28.6(i), 28.6(ii), 28.8(i), 28.8(ii)
Permalink:: http://dlmf.nist.gov/28.6.E26
Encodings:: pMML, png, TeX
See also:: Annotations for §28.6(ii), §28.6 and Ch.28

…

10: 27.2 Functions

… ►

Table 27.2.1: Primes.

► ►►►

$n$	$p_{n}$	$p_{n+10}$	$p_{n+20}$	$p_{n+30}$	$p_{n+40}$	$p_{n+50}$	$p_{n+60}$	$p_{n+70}$	$p_{n+80}$	$p_{n+90}$
…
9	23	67	109	167	227	277	347	401	461	523
…

► ►

Table 27.2.2: Functions related to division.

► ►►►►►►

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
…
5	4	2	6	18	6	6	39	31	30	2	32	44	20	6	84
6	2	4	12	19	18	2	20	32	16	6	63	45	24	6	78
…
8	4	4	15	21	12	4	32	34	16	4	54	47	46	2	48
…
12	4	6	28	25	20	3	31	38	18	4	60	51	32	4	72
…

►

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
…
5	4	2	6	18	6	6	39	31	30	2	32	44	20	6	84
6	2	4	12	19	18	2	20	32	16	6	63	45	24	6	78
…
8	4	4	15	21	12	4	32	34	16	4	54	47	46	2	48
…
12	4	6	28	25	20	3	31	38	18	4	60	51	32	4	72
…

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
…
5	4	2	6	18	6	6	39	31	30	2	32	44	20	6	84
6	2	4	12	19	18	2	20	32	16	6	63	45	24	6	78
…
8	4	4	15	21	12	4	32	34	16	4	54	47	46	2	48
…
12	4	6	28	25	20	3	31	38	18	4	60	51	32	4	72
…