34 3j, 6j, 9j Symbols Properties 34.5 Basic Properties:

\mathit{6j}

\mathit{9j}

Symbol

§34.6 Definition: $\mathit{9j}$ Symbol

ⓘ

Keywords:: definition, finite sum of $\mathit{3j}$ symbols, finite sum of $\mathit{6j}$ symbols, generalized hypergeometric functions, $\mathit{9j}$ symbols, representation as
Notes:: See Edmonds (1974, p. 101), de-Shalit and Talmi (1963, p. 516).
Permalink:: http://dlmf.nist.gov/34.6
See also:: Annotations for Ch.34

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	$\displaystyle\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}$	$\displaystyle=\sum_{\mbox{\scriptsize all }m_{rs}}\begin{pmatrix}j_{11}&j_{12}% &j_{13}\\ m_{11}&m_{12}&m_{13}\end{pmatrix}\begin{pmatrix}j_{21}&j_{22}&j_{23}\\ m_{21}&m_{22}&m_{23}\end{pmatrix}\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{31}&m_{32}&m_{33}\end{pmatrix}\*\begin{pmatrix}j_{11}&j_{21}&j_{31}\\ m_{11}&m_{21}&m_{31}\end{pmatrix}\begin{pmatrix}j_{12}&j_{22}&j_{32}\\ m_{12}&m_{22}&m_{32}\end{pmatrix}\begin{pmatrix}j_{13}&j_{23}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix},$
ⓘ Defines: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol Symbols: $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $r$ : nonnegative integer and $s$ : integer Permalink: http://dlmf.nist.gov/34.6.E1 Encodings: TeX, pMML, png See also: Annotations for §34.6 and Ch.34
34.6.2	$\displaystyle\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}$	$\displaystyle=\sum_{j}(-1)^{2j}(2j+1)\begin{Bmatrix}j_{11}&j_{21}&j_{31}\\ j_{32}&j_{33}&j\end{Bmatrix}\begin{Bmatrix}j_{12}&j_{22}&j_{32}\\ j_{21}&j&j_{23}\end{Bmatrix}\begin{Bmatrix}j_{13}&j_{23}&j_{33}\\ j&j_{11}&j_{12}\end{Bmatrix}.$
ⓘ Symbols: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol, $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol and $j,j_{r}$ : non-negative integers or non-negative integers plus one half. Referenced by: §34.13 Permalink: http://dlmf.nist.gov/34.6.E2 Encodings: TeX, pMML, png See also: Annotations for §34.6 and Ch.34

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. See Srinivasa Rao and Rajeswari (1993, pp. 7 and 125–132) and Rosengren (1999).

\mathit{6j}

Symbol 34.7 Basic Properties:

\mathit{9j}

Symbol

§34.6 Definition: 9⁢j Symbol

§34.6 Definition: $\mathit{9j}$ Symbol