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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 $\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{\mbox{\scriptsize all }m_{rs}}\begin{% pmatrix}j_{11}&j_{12}&j_{13}\\ m_{11}&m_{12}&m_{13}\end{pmatrix}\begin{pmatrix}j_{21}&j_{22}&j_{23}\\ m_{21}&m_{22}&m_{23}\end{pmatrix}\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{31}&m_{32}&m_{33}\end{pmatrix}\*\begin{pmatrix}j_{11}&j_{21}&j_{31}\\ m_{11}&m_{21}&m_{31}\end{pmatrix}\begin{pmatrix}j_{12}&j_{22}&j_{32}\\ m_{12}&m_{22}&m_{32}\end{pmatrix}\begin{pmatrix}j_{13}&j_{23}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix},$
34.6.2 $\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{j}(-1)^{2j}(2j+1)\begin{Bmatrix}j_{11}% &j_{21}&j_{31}\\ j_{32}&j_{33}&j\end{Bmatrix}\begin{Bmatrix}j_{12}&j_{22}&j_{32}\\ j_{21}&j&j_{23}\end{Bmatrix}\begin{Bmatrix}j_{13}&j_{23}&j_{33}\\ j&j_{11}&j_{12}\end{Bmatrix}.$
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. …
##### 3: Staff
• Frank W. J. Olver, University of Maryland and NIST, Chaps. 1, 2, 4, 9, 10

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

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

• Richard B. Paris, University of Abertay Dundee, for Chaps. 8, 11

• Hans Volkmer, University of Wisconsin–Milwaukee, for Chaps. 29, 30

• ##### 4: Publications
• B. R. Miller and A. Youssef (2003) Technical Aspects of the Digital Library of Mathematical Functions, Annals of Mathematics and Artificial Intelligence—Special Issue on Mathematical Knowledge Management, Vol. 38, Nos. 1–3, pp. 121–136. • B. V. Saunders and Q. Wang (2005) Boundary/Contour Fitted Grid Generation for Effective Visualizations in a Digital Library of Mathematical Functions, Proceedings of the 9th International Conference on Numerical Grid Generation in Computational Field Simulations, San Jose, June 11–18, 2005. pp. 61–71. • Q. Wang and B. V. Saunders (2005) Web-Based 3D Visualization in a Digital Library of Mathematical Functions, Proceedings of the Web3D Symposium, Bangor, UK, March 29–April 1, 2005. • B. V. Saunders and Q. Wang (2006) From B-Spline Mesh Generation to Effective Visualizations for the NIST Digital Library of Mathematical Functions, in Curve and Surface Design, Proceedings of the Sixth International Conference on Curves and Surfaces, Avignon, France June 29–July 5, 2006, pp. 235–243. ##### 6: 34.7 Basic Properties: $\mathit{9j}$ Symbol
###### §34.7(vi) Sums
It constitutes an addition theorem for the $\mathit{9j}$ symbol. …
##### 10: 34.9 Graphical Method
###### §34.9 Graphical Method
For specific examples of the graphical method of representing sums involving the $\mathit{3j},\mathit{6j}$, and $\mathit{9j}$ symbols, see Varshalovich et al. (1988, Chapters 11, 12) and Lehman and O’Connell (1973, §3.3).