# finite sum of 6j symbols

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##### 1: 34.6 Definition: $\mathit{9j}$ Symbol
βΊThe $\mathit{9j}$ symbol may be defined either in terms of $\mathit{3j}$ symbols or equivalently in terms of $\mathit{6j}$ symbols: βΊ
34.6.1 $\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{\mbox{\scriptsize all }m_{rs}}\begin{% pmatrix}j_{11}&j_{12}&j_{13}\\ m_{11}&m_{12}&m_{13}\end{pmatrix}\begin{pmatrix}j_{21}&j_{22}&j_{23}\\ m_{21}&m_{22}&m_{23}\end{pmatrix}\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{31}&m_{32}&m_{33}\end{pmatrix}\*\begin{pmatrix}j_{11}&j_{21}&j_{31}\\ m_{11}&m_{21}&m_{31}\end{pmatrix}\begin{pmatrix}j_{12}&j_{22}&j_{32}\\ m_{12}&m_{22}&m_{32}\end{pmatrix}\begin{pmatrix}j_{13}&j_{23}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix},$
##### 2: 34.4 Definition: $\mathit{6j}$ Symbol
###### §34.4 Definition: $\mathit{6j}$Symbol
βΊThe $\mathit{6j}$ symbol can be expressed as the finite sumβΊFor 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).
##### 3: 34.5 Basic Properties: $\mathit{6j}$ Symbol
βΊExamples are provided by: …
##### 4: Bibliography R
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• J. Raynal (1979) On the definition and properties of generalized $6$-$j$ symbols. J. Math. Phys. 20 (12), pp. 2398–2415.
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• H. Rosengren (1999) Another proof of the triple sum formula for Wigner $9j$-symbols. J. Math. Phys. 40 (12), pp. 6689–6691.
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• G. Rota, D. Kahaner, and A. Odlyzko (1973) On the foundations of combinatorial theory. VIII. Finite operator calculus. J. Math. Anal. Appl. 42, pp. 684–760.
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• M. Rotenberg, R. Bivins, N. Metropolis, and J. K. Wooten, Jr. (1959) The $3$-$j$ and $6$-$j$ Symbols. The Technology Press, MIT, Cambridge, MA.
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• K. Rottbrand (2000) Finite-sum rules for Macdonald’s functions and Hankel’s symbols. Integral Transform. Spec. Funct. 10 (2), pp. 115–124.
• ##### 5: Mathematical Introduction
βΊ βΊβΊβΊβΊβΊ
 $\mathbb{C}$ complex plane (excluding infinity). … zero. … double factorial: $2\cdot 4\cdot 6\cdots n$ if $n=2,4,6,\dotsc$; $1\cdot 3\cdot 5\cdots n$ if $n=1,3,5,\dotsc$; 1 if $n=0,-1$. … is finite, or converges. …
βΊ βΊβΊβΊ
 $(a,b]$ or $[a,b)$ half-closed intervals. … matrix with $(j,k)$th element $a_{j,k}$ or $a_{jk}$. …
βΊ J. …
##### 6: 10.22 Integrals
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###### §10.22(ii) Integrals over Finite Intervals
βΊWhen $\alpha=m=1,2,3,\ldots$ the left-hand side of (10.22.36) is the $m$th repeated integral of $J_{\nu}\left(x\right)$ (§§1.4(v) and 1.15(vi)). … βΊwhere $j_{\nu,\ell}$ and $j_{\nu,m}$ are zeros of $J_{\nu}\left(x\right)$10.21(i)), and $\delta_{\ell,m}$ is Kronecker’s symbol. … βΊwhere $\alpha_{\ell}$ and $\alpha_{m}$ are positive zeros of $aJ_{\nu}\left(x\right)+bxJ_{\nu}'\left(x\right)$. … …
##### 7: Errata
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• Subsection 31.11(iv)

Just below (31.11.17), $P_{j}$ has been replaced with $P_{j}^{6}$.

• βΊ
• The Legendre polynomial $P_{n}$ was mistakenly identified as the associated Legendre function $P_{n}$ in §§10.54, 10.59, 10.60, 18.18, 18.41, 34.3 (and was thus also affected by the bug reported below). These symbols now link correctly to their definitions. Reported by Roy Hughes on 2022-05-23

• βΊ
• Section 34.1

The relation between Clebsch-Gordan and $\mathit{3j}$ symbols was clarified, and the sign of $m_{3}$ was changed for readability. The reference Condon and Shortley (1935) for the Clebsch-Gordan coefficients was replaced by Edmonds (1974) and Rotenberg et al. (1959) and the references for $\mathit{3j}$, $\mathit{6j}$, $\mathit{9j}$ symbols were made more precise in §34.1.

• βΊ
• Section 34.1

The reference for Clebsch-Gordan coefficients, Condon and Shortley (1935), was replaced by Edmonds (1974) and Rotenberg et al. (1959). The references for $\mathit{3j}$, $\mathit{6j}$, $\mathit{9j}$ symbols were made more precise.

• βΊ
• Equation (34.7.4)
34.7.4 $\begin{pmatrix}j_{13}&j_{23}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix}\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{m_{r1},m_{r2},r=1,2,3}\begin{pmatrix}j% _{11}&j_{12}&j_{13}\\ m_{11}&m_{12}&m_{13}\end{pmatrix}\*\begin{pmatrix}j_{21}&j_{22}&j_{23}\\ m_{21}&m_{22}&m_{23}\end{pmatrix}\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{31}&m_{32}&m_{33}\end{pmatrix}\*\begin{pmatrix}j_{11}&j_{21}&j_{31}\\ m_{11}&m_{21}&m_{31}\end{pmatrix}\begin{pmatrix}j_{12}&j_{22}&j_{32}\\ m_{12}&m_{22}&m_{32}\end{pmatrix}$

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.

• ##### 8: Bibliography F
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• J. L. Fields and Y. L. Luke (1963b) Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. J. Math. Anal. Appl. 6 (3), pp. 394–403.
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• J. L. Fields (1983) Uniform asymptotic expansions of a class of Meijer $G$-functions for a large parameter. SIAM J. Math. Anal. 14 (6), pp. 1204–1253.
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• J. P. M. Flude (1998) The Edmonds asymptotic formulas for the $3j$ and $6j$ symbols. J. Math. Phys. 39 (7), pp. 3906–3915.
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• P. J. Forrester and N. S. Witte (2002) Application of the $\tau$-function theory of Painlevé equations to random matrices: $\mathrm{P}_{\mathrm{V}}$, $\mathrm{P}_{\mathrm{III}}$, the LUE, JUE, and CUE. Comm. Pure Appl. Math. 55 (6), pp. 679–727.
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• T. Fort (1948) Finite Differences and Difference Equations in the Real Domain. Clarendon Press, Oxford.
• ##### 9: Bibliography W
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• R. S. Ward (1987) The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.
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• G. N. Watson (1935b) The surface of an ellipsoid. Quart. J. Math., Oxford Ser. 6, pp. 280–287.
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• J. K. G. Watson (1999) Asymptotic approximations for certain $6$-$j$ and $9$-$j$ symbols. J. Phys. A 32 (39), pp. 6901–6902.
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• D. V. Widder (1941) The Laplace Transform. Princeton Mathematical Series, v. 6, Princeton University Press, Princeton, NJ.
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• C. Y. Wu (1982) A series of inequalities for Mills’s ratio. Acta Math. Sinica 25 (6), pp. 660–670.
• ##### 10: 25.11 Hurwitz Zeta Function
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25.11.10 $\zeta\left(s,a\right)=\sum_{n=0}^{\infty}\frac{{\left(s\right)_{n}}}{n!}\zeta% \left(n+s\right)(1-a)^{n},$ $s\neq 1$, $|a-1|<1$.
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25.11.36Removed because it is just (25.15.1) combined with (25.15.3).
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###### §25.11(xi) Sums
βΊFor further sums see Prudnikov et al. (1990, pp. 396–397) and Hansen (1975, pp. 358–360). … βΊ
25.11.43 $\zeta\left(s,a\right)-\frac{a^{1-s}}{s-1}-\frac{1}{2}a^{-s}\sim\sum_{k=1}^{% \infty}\frac{B_{2k}}{(2k)!}{\left(s\right)_{2k-1}}a^{1-s-2k}.$