§34.6 Definition: $9j$ Symbol

Notes:: See Edmonds (1974, p. 101), de-Shalit and Talmi (1963, p. 516).
Keywords:: generalized hypergeometric functions, $9j$ symbols
Permalink:: http://dlmf.nist.gov/34.6

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

34.6.1 $\begin{Bmatrix}j_{{11}}&j_{{12}}&j_{{13}}\\ j_{{21}}&j_{{22}}&j_{{23}}\\ j_{{31}}&j_{{32}}&j_{{33}}\end{Bmatrix}=\sum _{{\mbox{\scriptsize all }m_{{rs}}}}\begin{pmatrix}j_{{11}}&j_{{12}}&j_{{13}}\\ m_{{11}}&m_{{12}}&m_{{13}}\end{pmatrix}\begin{pmatrix}j_{{21}}&j_{{22}}&j_{{23}}\\ m_{{21}}&m_{{22}}&m_{{23}}\end{pmatrix}\begin{pmatrix}j_{{31}}&j_{{32}}&j_{{33}}\\ m_{{31}}&m_{{32}}&m_{{33}}\end{pmatrix}\*\begin{pmatrix}j_{{11}}&j_{{21}}&j_{{31}}\\ m_{{11}}&m_{{21}}&m_{{31}}\end{pmatrix}\begin{pmatrix}j_{{12}}&j_{{22}}&j_{{32}}\\ m_{{12}}&m_{{22}}&m_{{32}}\end{pmatrix}\begin{pmatrix}j_{{13}}&j_{{23}}&j_{{33}}\\ m_{{13}}&m_{{23}}&m_{{33}}\end{pmatrix},$

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 $\begin{Bmatrix}j_{{11}}&j_{{12}}&j_{{13}}\\ j_{{21}}&j_{{22}}&j_{{23}}\\ j_{{31}}&j_{{32}}&j_{{33}}\end{Bmatrix}=\sum _{{j}}(-1)^{{2j}}(2j+1)\begin{Bmatrix}j_{{11}}&j_{{21}}&j_{{31}}\\ j_{{32}}&j_{{33}}&j\end{Bmatrix}\begin{Bmatrix}j_{{12}}&j_{{22}}&j_{{32}}\\ j_{{21}}&j&j_{{23}}\end{Bmatrix}\begin{Bmatrix}j_{{13}}&j_{{23}}&j_{{33}}\\ j&j_{{11}}&j_{{12}}\end{Bmatrix}.$

Symbols:: $\begin{Bmatrix}j_{{11}}&j_{{12}}&j_{{13}}\\ j_{{21}}&j_{{22}}&j_{{23}}\\ j_{{31}}&j_{{32}}&j_{{33}}\end{Bmatrix}$ : $9j$ symbol, $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol and $j,j_{r}$ : nonnegative integer
Referenced by:: §34.13
Permalink:: http://dlmf.nist.gov/34.6.E2
Encodings:: TeX, pMML, png

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).

§34.6 Definition: Symbol

§34.6 Definition: $9j$ Symbol