# §34.3(i) Special Cases

When any one of $j_{1},j_{2},j_{3}$ is equal to $0,\tfrac{1}{2}$, or $1$, the $3j$ symbol has a simple algebraic form. Examples are provided by

 34.3.1 $\displaystyle\begin{pmatrix}j&j&0\\ m&-m&0\end{pmatrix}$ $\displaystyle=\frac{(-1)^{j-m}}{(2j+1)^{\frac{1}{2}}},$ 34.3.2 $\displaystyle\begin{pmatrix}j&j&1\\ m&-m&0\end{pmatrix}$ $\displaystyle=(-1)^{j-m}\frac{2m}{\left(2j(2j+1)(2j+2)\right)^{\frac{1}{2}}},$ $j\geq\tfrac{1}{2}$, 34.3.3 $\displaystyle\begin{pmatrix}j&j&1\\ m&-m-1&1\end{pmatrix}$ $\displaystyle=(-1)^{j-m}\left(\frac{2(j-m)(j+m+1)}{2j(2j+1)(2j+2)}\right)^{% \frac{1}{2}},$ $j\geq\tfrac{1}{2}$.

For these and other results, and also cases in which any one of $j_{1},j_{2},j_{3}$ is $\frac{3}{2}$ or $2$, see Edmonds (1974, pp. 125–127).

Next define

 34.3.4 $J=j_{1}+j_{2}+j_{3}.$ Symbols: $j,j_{r}$: nonnegative integer and $J$: sum Referenced by: §34.5(i) Permalink: http://dlmf.nist.gov/34.3.E4 Encodings: TeX, pMML, png

Then assuming the triangle conditions are satisfied

 34.3.5 $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ 0&0&0\end{pmatrix}=\begin{cases}0,&\mbox{J odd},\\ (-1)^{\frac{1}{2}J}\left(\dfrac{(J-2j_{1})!(J-2j_{2})!(J-2j_{3})!}{(J+1)!}% \right)^{\frac{1}{2}}\dfrac{(\frac{1}{2}J)!}{(\frac{1}{2}J-j_{1})!(\frac{1}{2}% J-j_{2})!(\frac{1}{2}J-j_{3})!},&\mbox{J even}.\end{cases}$

Lastly,

 34.3.6 $\begin{pmatrix}j_{1}&j_{2}&j_{1}+j_{2}\\ m_{1}&m_{2}&-m_{1}-m_{2}\end{pmatrix}=(-1)^{j_{1}-j_{2}+m_{1}+m_{2}}\left(% \frac{(2j_{1})!(2j_{2})!(j_{1}+j_{2}+m_{1}+m_{2})!(j_{1}+j_{2}-m_{1}-m_{2})!}{% (2j_{1}+2j_{2}+1)!(j_{1}+m_{1})!(j_{1}-m_{1})!(j_{2}+m_{2})!(j_{2}-m_{2})!}% \right)^{\frac{1}{2}},$
 34.3.7 $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ j_{1}&-j_{1}-m_{3}&m_{3}\end{pmatrix}=(-1)^{j_{1}-j_{2}-m_{3}}\left(\frac{(2j_% {1})!(-j_{1}+j_{2}+j_{3})!(j_{1}+j_{2}+m_{3})!(j_{3}-m_{3})!}{(j_{1}+j_{2}+j_{% 3}+1)!(j_{1}-j_{2}+j_{3})!(j_{1}+j_{2}-j_{3})!(-j_{1}+j_{2}-m_{3})!(j_{3}+m_{3% })!}\right)^{\frac{1}{2}}.$ Symbols: $!$: $n!$: factorial, $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$: $3j$ symbol and $j,j_{r}$: nonnegative integer Referenced by: §34.3(i), Equation (34.3.7) Permalink: http://dlmf.nist.gov/34.3.E7 Encodings: TeX, pMML, png Errata (effective with 1.0.9): The prefactor $(-1)^{-j_{2}+j_{3}+m_{3}}$ in the above 3j symbol was replaced by its correct value $(-1)^{j_{1}-j_{2}-m_{3}}$ Reported 2014-06-12 by James Zibin

Again it is assumed that in (34.3.7) the triangle conditions are satisfied.

# §34.3(ii) Symmetry

Even permutations of columns of a $3j$ symbol leave it unchanged; odd permutations of columns produce a phase factor $(-1)^{j_{1}+j_{2}+j_{3}}$, for example,

 34.3.8 $\displaystyle\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$ $\displaystyle=\begin{pmatrix}j_{2}&j_{3}&j_{1}\\ m_{2}&m_{3}&m_{1}\end{pmatrix}$ $\displaystyle=\begin{pmatrix}j_{3}&j_{1}&j_{2}\\ m_{3}&m_{1}&m_{2}\end{pmatrix},$ 34.3.9 $\displaystyle\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$ $\displaystyle=(-1)^{j_{1}+j_{2}+j_{3}}\begin{pmatrix}j_{2}&j_{1}&j_{3}\\ m_{2}&m_{1}&m_{3}\end{pmatrix}.$

Next,

 34.3.10 $\displaystyle\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$ $\displaystyle=(-1)^{j_{1}+j_{2}+j_{3}}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ -m_{1}&-m_{2}&-m_{3}\end{pmatrix},$ 34.3.11 $\displaystyle\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$ $\displaystyle=\begin{pmatrix}j_{1}&\frac{1}{2}(j_{2}+j_{3}+m_{1})&\frac{1}{2}(% j_{2}+j_{3}-m_{1})\\ j_{2}-j_{3}&\frac{1}{2}(j_{3}-j_{2}+m_{1})+m_{2}&\frac{1}{2}(j_{3}-j_{2}+m_{1}% )+m_{3}\end{pmatrix},$ 34.3.12 $\displaystyle\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$ $\displaystyle=\begin{pmatrix}\frac{1}{2}(j_{1}+j_{2}-m_{3})&\frac{1}{2}(j_{2}+% j_{3}-m_{1})&\frac{1}{2}(j_{1}+j_{3}-m_{2})\\ j_{3}-\frac{1}{2}(j_{1}+j_{2}+m_{3})&j_{1}-\frac{1}{2}(j_{2}+j_{3}+m_{1})&j_{2% }-\frac{1}{2}(j_{1}+j_{3}+m_{2})\end{pmatrix}.$

Equations (34.3.11) and (34.3.12) are called Regge symmetries. Additional symmetries are obtained by applying (34.3.8)–(34.3.10) to (34.3.11)) and (34.3.12). See Srinivasa Rao and Rajeswari (1993, pp. 44–47) and references given there.

# §34.3(iii) Recursion Relations

In the following three equations it is assumed that the triangle conditions are satisfied by each $3j$ symbol.

 34.3.13 $\left((j_{1}+j_{2}+j_{3}+1)(-j_{1}+j_{2}+j_{3})\right)^{\frac{1}{2}}\begin{% pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}=\left((j_{2}+m_{2})(j_{3}-m_{3})\right)^{\frac{% 1}{2}}\begin{pmatrix}j_{1}&j_{2}-\frac{1}{2}&j_{3}-\frac{1}{2}\\ m_{1}&m_{2}-\frac{1}{2}&m_{3}+\frac{1}{2}\end{pmatrix}-\left((j_{2}-m_{2})(j_{% 3}+m_{3})\right)^{\frac{1}{2}}\begin{pmatrix}j_{1}&j_{2}-\frac{1}{2}&j_{3}-% \frac{1}{2}\\ m_{1}&m_{2}+\frac{1}{2}&m_{3}-\frac{1}{2}\end{pmatrix},$
 34.3.14 $\left(j_{1}(j_{1}+1)-j_{2}(j_{2}+1)-j_{3}(j_{3}+1)-2m_{2}m_{3}\right)\begin{% pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}=\left((j_{2}-m_{2})(j_{2}+m_{2}+1)(j_{3}-m_{3}+% 1)(j_{3}+m_{3})\right)^{\frac{1}{2}}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}+1&m_{3}-1\end{pmatrix}+\left((j_{2}-m_{2}+1)(j_{2}+m_{2})(j_{3}-m_% {3})(j_{3}+m_{3}+1)\right)^{\frac{1}{2}}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}-1&m_{3}+1\end{pmatrix},$
 34.3.15 $(2j_{1}+1)\left((j_{2}(j_{2}+1)-j_{3}(j_{3}+1))m_{1}-j_{1}(j_{1}+1)(m_{3}-m_{2% })\right)\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}=(j_{1}+1)\left(j_{1}^{2}-(j_{2}-j_{3})^{2}% \right)^{\frac{1}{2}}\left((j_{2}+j_{3}+1)^{2}-j_{1}^{2}\right)^{\frac{1}{2}}% \left(j_{1}^{2}-m_{1}^{2}\right)^{\frac{1}{2}}\begin{pmatrix}j_{1}-1&j_{2}&j_{% 3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}+j_{1}\left((j_{1}+1)^{2}-(j_{2}-j_{3})^{2}% \right)^{\frac{1}{2}}\left((j_{2}+j_{3}+1)^{2}-(j_{1}+1)^{2}\right)^{\frac{1}{% 2}}\left((j_{1}+1)^{2}-m_{1}^{2}\right)^{\frac{1}{2}}\begin{pmatrix}j_{1}+1&j_% {2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}.$

For these and other recursion relations see Varshalovich et al. (1988, §8.6). See also Micu (1968), Louck (1958), Schulten and Gordon (1975a), Srinivasa Rao and Rajeswari (1993, pp. 220–225), and Luscombe and Luban (1998).

# §34.3(iv) Orthogonality

 34.3.16 $\displaystyle\sum_{m_{1}m_{2}}(2j_{3}+1)\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}\begin{pmatrix}j_{1}&j_{2}&j^{\prime}_{3}\\ m_{1}&m_{2}&m^{\prime}_{3}\end{pmatrix}$ $\displaystyle=\delta_{j_{3},j^{\prime}_{3}}\delta_{m_{3},m^{\prime}_{3}},$ 34.3.17 $\displaystyle\sum_{j_{3}m_{3}}(2j_{3}+1)\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m^{\prime}_{1}&m^{\prime}_{2}&m_{3}\end{pmatrix}$ $\displaystyle=\delta_{m_{1},m^{\prime}_{1}}\delta_{m_{2},m^{\prime}_{2}},$ 34.3.18 $\displaystyle\sum_{m_{1}m_{2}m_{3}}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$ $\displaystyle=1.$

In the summations (34.3.16)–(34.3.18) the summation variables range over all values that satisfy the conditions given in (34.2.1)–(34.2.3). Similar conventions apply to all subsequent summations in this chapter.

# §34.3(v) Generating Functions

For generating functions for the $3j$ symbol see Biedenharn and van Dam (1965, p. 245, Eq. (3.42) and p. 247, Eq. (3.55)).

# §34.3(vi) Sums

For sums of products of $3j$ symbols, see Varshalovich et al. (1988, pp. 259–262).

# §34.3(vii) Relations to Legendre Polynomials and Spherical Harmonics

For the polynomials $\mathop{P_{l}\/}\nolimits$ see §18.3, and for the functions $\mathop{Y_{{l},{m}}\/}\nolimits$ and ${\mathop{Y_{{l},{m}}\/}\nolimits^{*}}$ see §14.30.

 34.3.19 $\mathop{P_{l_{1}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta\right)% \mathop{P_{l_{2}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta\right)=% \sum_{l}(2l+1)\begin{pmatrix}l_{1}&l_{2}&l\\ 0&0&0\end{pmatrix}^{2}\mathop{P_{l}\/}\nolimits\!\left(\mathop{\cos\/}% \nolimits\theta\right),$
 34.3.20 $\mathop{Y_{{l_{1}},{m_{1}}}\/}\nolimits\!\left(\theta,\phi\right)\mathop{Y_{{l% _{2}},{m_{2}}}\/}\nolimits\!\left(\theta,\phi\right)=\sum_{l,m}\left(\frac{(2l% _{1}+1)(2l_{2}+1)(2l+1)}{4\pi}\right)^{\frac{1}{2}}\begin{pmatrix}l_{1}&l_{2}&% l\\ m_{1}&m_{2}&m\end{pmatrix}{\mathop{Y_{{l},{m}}\/}\nolimits^{*}}\!\left(\theta,% \phi\right)\begin{pmatrix}l_{1}&l_{2}&l\\ 0&0&0\end{pmatrix},$
 34.3.21 $\int_{0}^{\pi}\mathop{P_{l_{1}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits% \theta\right)\mathop{P_{l_{2}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits% \theta\right)\mathop{P_{l_{3}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits% \theta\right)\mathop{\sin\/}\nolimits\theta d\theta=2\begin{pmatrix}l_{1}&l_{2% }&l_{3}\\ 0&0&0\end{pmatrix}^{2},$
 34.3.22 $\int_{0}^{2\pi}\!\int_{0}^{\pi}\mathop{Y_{{l_{1}},{m_{1}}}\/}\nolimits\!\left(% \theta,\phi\right)\mathop{Y_{{l_{2}},{m_{2}}}\/}\nolimits\!\left(\theta,\phi% \right)\mathop{Y_{{l_{3}},{m_{3}}}\/}\nolimits\!\left(\theta,\phi\right)% \mathop{\sin\/}\nolimits\theta d\theta d\phi=\left(\frac{(2l_{1}+1)(2l_{2}+1)(% 2l_{3}+1)}{4\pi}\right)^{\frac{1}{2}}\begin{pmatrix}l_{1}&l_{2}&l_{3}\\ 0&0&0\end{pmatrix}\begin{pmatrix}l_{1}&l_{2}&l_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}.$

Equations (34.3.19)–(34.3.22) are particular cases of more general results that relate rotation matrices to $3j$ symbols, for which see Edmonds (1974, Chapter 4). The left- and right-hand sides of (34.3.22) are known, respectively, as Gaunt’s integral and the Gaunt coefficient (Gaunt (1929)).