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1: 32 Painlevé Transcendents

Chapter 32 Painlevé Transcendents

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2: 34.6 Definition: $\mathit{9j}$ Symbol

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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},

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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

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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}.

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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

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3: Staff

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

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Richard B. Paris, University of Abertay, Chaps. 8, 11

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Hans Volkmer, University of Wisconsin, Milwaukee, Chaps. 29, 30

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

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Richard B. Paris, University of Abertay Dundee, for Chaps. 8, 11 (deceased)

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4: 34.7 Basic Properties: $\mathit{9j}$ Symbol

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34.7.1

\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{13}\\ j_{31}&j_{31}&0\end{Bmatrix}=\frac{(-1)^{j_{12}+j_{21}+j_{13}+j_{31}}}{((2j_{1% 3}+1)(2j_{31}+1))^{\frac{1}{2}}}\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{22}&j_{21}&j_{31}\end{Bmatrix}.

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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.
Permalink:: http://dlmf.nist.gov/34.7.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §34.7(i), §34.7 and Ch.34

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34.7.2

\sum_{j_{12}\,j_{34}}(2j_{12}+1)(2j_{34}+1)(2j_{13}+1)(2j_{24}+1)\begin{% Bmatrix}j_{1}&j_{2}&j_{12}\\ j_{3}&j_{4}&j_{34}\\ j_{13}&j_{24}&j\end{Bmatrix}\begin{Bmatrix}j_{1}&j_{2}&j_{12}\\ j_{3}&j_{4}&j_{34}\\ j^{\prime}_{13}&j^{\prime}_{24}&j\end{Bmatrix}=\delta_{j_{13},j^{\prime}_{13}}% \delta_{j_{24},j^{\prime}_{24}}.

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Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol and $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Permalink:: http://dlmf.nist.gov/34.7.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §34.7(iv), §34.7 and Ch.34

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34.7.3

\sum_{j_{13}\,j_{24}}(-1)^{2j_{2}+j_{24}+j_{23}-j_{34}}(2j_{13}+1)(2j_{24}+1)% \begin{Bmatrix}j_{1}&j_{2}&j_{12}\\ j_{3}&j_{4}&j_{34}\\ j_{13}&j_{24}&j\end{Bmatrix}\begin{Bmatrix}j_{1}&j_{3}&j_{13}\\ j_{4}&j_{2}&j_{24}\\ j_{14}&j_{23}&j\end{Bmatrix}=\begin{Bmatrix}j_{1}&j_{2}&j_{12}\\ j_{4}&j_{3}&j_{34}\\ j_{14}&j_{23}&j\end{Bmatrix}.

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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 and $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Referenced by:: §34.9
Permalink:: http://dlmf.nist.gov/34.7.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §34.7(vi), §34.7 and Ch.34

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34.7.4

\begin{pmatrix}j_{13}&j_{23}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix}\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{m_{r1},m_{r2},r=1,2,3}\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}.

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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, $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $r$ : nonnegative integer
Referenced by:: Erratum (V1.0.10) for Equation (34.7.4)
Permalink:: http://dlmf.nist.gov/34.7.E4
Encodings:: pMML, png, TeX
Errata (effective with 1.0.10):: Originally the third $\mathit{3j}$ symbol in the summation was written incorrectly as $\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix}.$
Reported 2015-01-19 by Yan-Rui Liu
See also:: Annotations for §34.7(vi), §34.7 and Ch.34

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34.7.5

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

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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.
Permalink:: http://dlmf.nist.gov/34.7.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §34.7(vi), §34.7 and Ch.34

5: 26.2 Basic Definitions

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Table 26.2.1: Partitions

p\left(n\right)

.

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$n$	$p\left(n\right)$	$n$	$p\left(n\right)$	$n$	$p\left(n\right)$
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6	11	23	1255	40	37338
7	15	24	1575	41	44583
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11	56	28	3718	45	89134
12	77	29	4565	46	1 05558
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15	176	32	8349	49	1 73525
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6: Bibliography E

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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.

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

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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.

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

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

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

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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.

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

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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.

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

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7: 27.2 Functions

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Table 27.2.1: Primes.

$n$	$p_{n}$	$p_{n+10}$	$p_{n+20}$	$p_{n+30}$	$p_{n+40}$	$p_{n+50}$	$p_{n+60}$	$p_{n+70}$	$p_{n+80}$	$p_{n+90}$
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5	11	47	97	149	197	257	313	379	439	499
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Table 27.2.2: Functions related to division.

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$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
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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
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8	4	4	15	21	12	4	32	34	16	4	54	47	46	2	48
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12	4	6	28	25	20	3	31	38	18	4	60	51	32	4	72
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8: 4.17 Special Values and Limits

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Table 4.17.1: Trigonometric functions: values at multiples of

\frac{1}{12}\pi

.

$\theta$	$\sin\theta$	$\cos\theta$	$\tan\theta$	$\csc\theta$	$\sec\theta$	$\cot\theta$
…
$11\pi/12$	$\frac{1}{4}\sqrt{2}(\sqrt{3}-1)$	$-\frac{1}{4}\sqrt{2}(\sqrt{3}+1)$	$-(2-\sqrt{3})$	$\sqrt{2}(\sqrt{3}+1)$	$-\sqrt{2}(\sqrt{3}-1)$	$-(2+\sqrt{3})$
…

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9: 28.15 Expansions for Small $q$

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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.

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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

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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;

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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

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10: 1.3 Determinants, Linear Operators, and Spectral Expansions

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1.3.1

\det[a_{jk}]=\begin{vmatrix}a_{11}&a_{12}\\ a_{21}&a_{22}\end{vmatrix}=a_{11}a_{22}-a_{12}a_{21}.

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Symbols:: $\det$ : determinant, $j$ : integer and $k$ : integer
Permalink:: http://dlmf.nist.gov/1.3.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §1.3(i), §1.3(i), §1.3 and Ch.1

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1.3.2

\det[a_{jk}]=\begin{vmatrix}a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\end{vmatrix}=a_{11}\begin{vmatrix}a_{22}&a_{23}\\ a_{32}&a_{33}\end{vmatrix}-a_{12}\begin{vmatrix}a_{21}&a_{23}\\ a_{31}&a_{33}\end{vmatrix}+a_{13}\begin{vmatrix}a_{21}&a_{22}\\ a_{31}&a_{32}\end{vmatrix}=a_{11}a_{22}a_{33}-a_{11}a_{23}a_{32}-a_{12}a_{21}a% _{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}-a_{13}a_{22}a_{31}.

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Symbols:: $\det$ : determinant, $j$ : integer and $k$ : integer
Permalink:: http://dlmf.nist.gov/1.3.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §1.3(i), §1.3(i), §1.3 and Ch.1

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1.3.8

{\begin{vmatrix}a_{11}&a_{12}\\ a_{21}&a_{22}\end{vmatrix}}^{2}\leq(a^{2}_{11}+a^{2}_{12})(a^{2}_{21}+a^{2}_{2% 2}),

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Symbols:: $\det$ : determinant
Permalink:: http://dlmf.nist.gov/1.3.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §1.3(i), §1.3(i), §1.3 and Ch.1

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