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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. …
##### 2: 19.15 Advantages of Symmetry
Elliptic integrals are special cases of a particular multivariate hypergeometric function called Lauricella’s $F_{D}$ (Carlson (1961b)). The function $R_{-a}\left(b_{1},b_{2},\dots,b_{n};z_{1},z_{2},\dots,z_{n}\right)$ (Carlson (1963)) reveals the full permutation symmetry that is partially hidden in $F_{D}$, and leads to symmetric standard integrals that simplify many aspects of theory, applications, and numerical computation. …
##### 3: 34.12 Physical Applications
###### §34.12 Physical Applications
The angular momentum coupling coefficients ($\mathit{3j}$, $\mathit{6j}$, and $\mathit{9j}$ symbols) are essential in the fields of nuclear, atomic, and molecular physics. …$\mathit{3j},\mathit{6j}$, and $\mathit{9j}$ symbols are also found in multipole expansions of solutions of the Laplace and Helmholtz equations; see Carlson and Rushbrooke (1950) and Judd (1976).
##### 4: Bibliography F
• V. Fock (1945) Diffraction of radio waves around the earth’s surface. Acad. Sci. USSR. J. Phys. 9, pp. 255–266.
• B. R. Frieden (1971) Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions. In Progress in Optics, E. Wolf (Ed.), Vol. 9, pp. 311–407.
• L. W. Fullerton and G. A. Rinker (1986) Generalized Fermi-Dirac integrals—FD, FDG, FDH. Comput. Phys. Comm. 39 (2), pp. 181–185.
##### 7: 34.14 Tables
###### §34.14 Tables
Tables of exact values of the squares of the $\mathit{3j}$ and $\mathit{6j}$ symbols in which all parameters are $\leq 8$ are given in Rotenberg et al. (1959), together with a bibliography of earlier tables of $\mathit{3j},\mathit{6j}$, and $\mathit{9j}$ symbols on pp. … Some selected $\mathit{9j}$ symbols are also given. … 16-17; for $\mathit{9j}$ symbols on p. …  310–332, and for the $\mathit{9j}$ symbols on pp. …
##### 8: 19.25 Relations to Other Functions
For analogous integrals of the second kind, which are not invertible in terms of single-valued functions, see (19.29.20) and (19.29.21) and compare with Gradshteyn and Ryzhik (2000, §3.153,1–10 and §3.156,1–9). … For these results and extensions to the Appell function ${F_{1}}$16.13) and Lauricella’s function $F_{D}$ see Carlson (1963). (${F_{1}}$ and $F_{D}$ are equivalent to the $R$-function of 3 and $n$ variables, respectively, but lack full symmetry.) …
##### 9: 16.26 Approximations
For discussions of the approximation of generalized hypergeometric functions and the Meijer $G$-function in terms of polynomials, rational functions, and Chebyshev polynomials see Luke (1975, §§5.12 - 5.13) and Luke (1977b, Chapters 1 and 9).
##### 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).