# ninej symbols

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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: 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).
##### 3: 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. …
##### 4: 34.13 Methods of Computation
Methods of computation for $\mathit{3j}$ and $\mathit{6j}$ symbols include recursion relations, see Schulten and Gordon (1975a), Luscombe and Luban (1998), and Edmonds (1974, pp. 42–45, 48–51, 97–99); summation of single-sum expressions for these symbols, see Varshalovich et al. (1988, §§8.2.6, 9.2.1) and Fang and Shriner (1992); evaluation of the generalized hypergeometric functions of unit argument that represent these symbols, see Srinivasa Rao and Venkatesh (1978) and Srinivasa Rao (1981). For $\mathit{9j}$ symbols, methods include evaluation of the single-sum series (34.6.2), see Fang and Shriner (1992); evaluation of triple-sum series, see Varshalovich et al. (1988, §10.2.1) and Srinivasa Rao et al. (1989). …
##### 5: 34.10 Zeros
Such zeros are called nontrivial zeros. For further information, including examples of nontrivial zeros and extensions to $\mathit{9j}$ symbols, see Srinivasa Rao and Rajeswari (1993, pp. 133–215, 294–295, 299–310).
##### 6: 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).
##### 7: 34.7 Basic Properties: $\mathit{9j}$ Symbol
###### §34.7(vi) Sums
It constitutes an addition theorem for the $\mathit{9j}$ symbol. …
##### 8: 16.24 Physical Applications
###### §16.24(iii) $\mathit{3j}$, $\mathit{6j}$, and $\mathit{9j}$Symbols
Lastly, special cases of the $\mathit{9j}$ symbols are ${{}_{5}F_{4}}$ functions with unit argument. …
##### 9: 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, … 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).
##### 10: 34.8 Approximations for Large Parameters
###### §34.8 Approximations for Large Parameters
For large values of the parameters in the $\mathit{3j}$, $\mathit{6j}$, and $\mathit{9j}$ symbols, different asymptotic forms are obtained depending on which parameters are large. … For approximations for the $\mathit{3j}$, $\mathit{6j}$, and $\mathit{9j}$ symbols with error bounds see Flude (1998), Chen et al. (1999), and Watson (1999): these references also cite earlier work.