# §34.6 Definition: $9j$ Symbol

The $9j$ symbol may be defined either in terms of $3j$ symbols or equivalently in terms of $6j$ symbols:

 34.6.1 $\displaystyle\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}$ $\displaystyle=\sum_{\mbox{\scriptsize all }m_{rs}}\begin{pmatrix}j_{11}&j_{12}% &j_{13}\\ m_{11}&m_{12}&m_{13}\end{pmatrix}\begin{pmatrix}j_{21}&j_{22}&j_{23}\\ m_{21}&m_{22}&m_{23}\end{pmatrix}\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{31}&m_{32}&m_{33}\end{pmatrix}\*\begin{pmatrix}j_{11}&j_{21}&j_{31}\\ m_{11}&m_{21}&m_{31}\end{pmatrix}\begin{pmatrix}j_{12}&j_{22}&j_{32}\\ m_{12}&m_{22}&m_{32}\end{pmatrix}\begin{pmatrix}j_{13}&j_{23}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix},$ Defines: $\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}$: $9j$ symbol Symbols: $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$: $3j$ symbol, $j,j_{r}$: nonnegative integer, $r$: nonnegative integer and $s$: integer Permalink: http://dlmf.nist.gov/34.6.E1 Encodings: TeX, pMML, png 34.6.2 $\displaystyle\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}$ $\displaystyle=\sum_{j}(-1)^{2j}(2j+1)\begin{Bmatrix}j_{11}&j_{21}&j_{31}\\ j_{32}&j_{33}&j\end{Bmatrix}\begin{Bmatrix}j_{12}&j_{22}&j_{32}\\ j_{21}&j&j_{23}\end{Bmatrix}\begin{Bmatrix}j_{13}&j_{23}&j_{33}\\ j&j_{11}&j_{12}\end{Bmatrix}.$

The $9j$ symbol may also be written as a finite triple sum equivalent to a terminating generalized hypergeometric series of three variables with unit arguments. See Srinivasa Rao and Rajeswari (1993, pp. 7 and 125–132) and Rosengren (1999).