# §34.1 Special Notation

(For other notation see Notation for the Special Functions.)

$2j_{1},2j_{2},2j_{3},2l_{1},2l_{2},2l_{3}$ nonnegative integers. nonnegative integers.

The main functions treated in this chapter are the Wigner $\mathit{3j},\mathit{6j},\mathit{9j}$ symbols, respectively,

 $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix},$ $\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix},$ $\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}.$

The most commonly used alternative notation for the $\mathit{3j}$ symbol is the Clebsch–Gordan coefficient

 $\left(j_{1}m_{1}j_{2}m_{2}|j_{1}j_{2}j_{3}-m_{3}\right)=(-1)^{j_{1}-j_{2}-m_{3% }}(2j_{3}+1)^{\frac{1}{2}}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix};$

see Condon and Shortley (1935). For other notations see Edmonds (1974, pp. 52, 97, 104–105) and Varshalovich et al. (1988, §§8.11, 9.10, 10.10).