34.5 Basic Properties: 6j Symbol34.7 Basic Properties: 9j Symbol

§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\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 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).