34 3j, 6j, 9j Symbols Properties 34.4 Definition:

\mathit{6j}

\mathit{9j}

Symbol

§34.5 Basic Properties: $\mathit{6j}$ Symbol

ⓘ

Permalink:: http://dlmf.nist.gov/34.5
See also:: Annotations for Ch.34

§34.5(i) Special Cases
§34.5(ii) Symmetry
§34.5(iii) Recursion Relations
§34.5(iv) Orthogonality
§34.5(v) Generating Functions
§34.5(vi) Sums

§34.5(i) Special Cases

ⓘ

Keywords:: finite sum of algebraic quantities, representation as, $\mathit{6j}$ symbols, special cases
Notes:: See Edmonds (1974, pp. 98, 130–132), de-Shalit and Talmi (1963, p. 520).
Permalink:: http://dlmf.nist.gov/34.5.i
See also:: Annotations for §34.5 and Ch.34

In the following equations it is assumed that the triangle inequalities are satisfied and that $J$ is again defined by (34.3.4).

If any lower argument in a $\mathit{6j}$ symbol is $0$ , $\tfrac{1}{2}$ , or $1$ , then the $\mathit{6j}$ symbol has a simple algebraic form. Examples are provided by:

34.5.1	$\displaystyle\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ 0&j_{3}&j_{2}\end{Bmatrix}$	$\displaystyle=\frac{(-1)^{J}}{\left((2j_{2}+1)(2j_{3}+1)\right)^{\frac{1}{2}}},$
ⓘ Symbols: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $J$ : sum Permalink: http://dlmf.nist.gov/34.5.E1 Encodings: TeX, pMML, png See also: Annotations for §34.5(i), §34.5 and Ch.34
34.5.2	$\displaystyle\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ \frac{1}{2}&j_{3}-\frac{1}{2}&j_{2}+\frac{1}{2}\end{Bmatrix}$	$\displaystyle=(-1)^{J}\left(\frac{(j_{1}+j_{3}-j_{2})(j_{1}+j_{2}-j_{3}+1)}{(2% j_{2}+1)(2j_{2}+2)2j_{3}(2j_{3}+1)}\right)^{\frac{1}{2}},$
ⓘ Symbols: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $J$ : sum Permalink: http://dlmf.nist.gov/34.5.E2 Encodings: TeX, pMML, png See also: Annotations for §34.5(i), §34.5 and Ch.34
34.5.3	$\displaystyle\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ \frac{1}{2}&j_{3}-\frac{1}{2}&j_{2}-\frac{1}{2}\end{Bmatrix}$	$\displaystyle=(-1)^{J}\left(\frac{(j_{2}+j_{3}-j_{1})(j_{1}+j_{2}+j_{3}+1)}{2j% _{2}(2j_{2}+1)2j_{3}(2j_{3}+1)}\right)^{\frac{1}{2}},$
ⓘ Symbols: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $J$ : sum Permalink: http://dlmf.nist.gov/34.5.E3 Encodings: TeX, pMML, png See also: Annotations for §34.5(i), §34.5 and Ch.34
34.5.4	$\displaystyle\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ 1&j_{3}-1&j_{2}-1\end{Bmatrix}$	$\displaystyle=(-1)^{J}\left(\frac{J(J+1)(J-2j_{1})(J-2j_{1}-1)}{(2j_{2}-1)2j_{% 2}(2j_{2}+1)(2j_{3}-1)2j_{3}(2j_{3}+1)}\right)^{\frac{1}{2}},$
ⓘ Symbols: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $J$ : sum Permalink: http://dlmf.nist.gov/34.5.E4 Encodings: TeX, pMML, png See also: Annotations for §34.5(i), §34.5 and Ch.34
34.5.5	$\displaystyle\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ 1&j_{3}-1&j_{2}\end{Bmatrix}$	$\displaystyle=(-1)^{J}\left(\frac{2(J+1)(J-2j_{1})(J-2j_{2})(J-2j_{3}+1)}{2j_{% 2}(2j_{2}+1)(2j_{2}+2)(2j_{3}-1)2j_{3}(2j_{3}+1)}\right)^{\frac{1}{2}},$
ⓘ Symbols: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $J$ : sum Permalink: http://dlmf.nist.gov/34.5.E5 Encodings: TeX, pMML, png See also: Annotations for §34.5(i), §34.5 and Ch.34
34.5.6	$\displaystyle\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ 1&j_{3}-1&j_{2}+1\end{Bmatrix}$	$\displaystyle=(-1)^{J}\left(\frac{(J-2j_{2}-1)(J-2j_{2})(J-2j_{3}+1)(J-2j_{3}+% 2)}{(2j_{2}+1)(2j_{2}+2)(2j_{2}+3)(2j_{3}-1)2j_{3}(2j_{3}+1)}\right)^{\frac{1}% {2}},$
ⓘ Symbols: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $J$ : sum Permalink: http://dlmf.nist.gov/34.5.E6 Encodings: TeX, pMML, png See also: Annotations for §34.5(i), §34.5 and Ch.34
34.5.7	$\displaystyle\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ 1&j_{3}&j_{2}\end{Bmatrix}$	$\displaystyle=(-1)^{J+1}\frac{2(j_{2}(j_{2}+1)+j_{3}(j_{3}+1)-j_{1}(j_{1}+1))}% {\left(2j_{2}(2j_{2}+1)(2j_{2}+2)2j_{3}(2j_{3}+1)(2j_{3}+2)\right)^{\frac{1}{2% }}}.$
ⓘ Symbols: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $J$ : sum Permalink: http://dlmf.nist.gov/34.5.E7 Encodings: TeX, pMML, png See also: Annotations for §34.5(i), §34.5 and Ch.34

§34.5(ii) Symmetry

ⓘ

Keywords:: $\mathit{6j}$ symbols, Regge symmetries, $\mathit{6j}$ symbols, symmetry
Notes:: See Edmonds (1974, pp. 94, 95).
Permalink:: http://dlmf.nist.gov/34.5.ii
See also:: Annotations for §34.5 and Ch.34

The $\mathit{6j}$ symbol is invariant under interchange of any two columns and also under interchange of the upper and lower arguments in each of any two columns, for example,

34.5.8		$\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}=\begin{Bmatrix}j_{2}&j_{1}&j_{3}\\ l_{2}&l_{1}&l_{3}\end{Bmatrix}=\begin{Bmatrix}j_{1}&l_{2}&l_{3}\\ l_{1}&j_{2}&j_{3}\end{Bmatrix}.$
ⓘ Symbols: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $l,l_{r}$ : non-negative integers or non-negative integers plus one half. Referenced by: §34.5(ii) Permalink: http://dlmf.nist.gov/34.5.E8 Encodings: TeX, pMML, png See also: Annotations for §34.5(ii), §34.5 and Ch.34

Next,

34.5.9	$\displaystyle\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}$	$\displaystyle=\begin{Bmatrix}j_{1}&\frac{1}{2}(j_{2}+l_{2}+j_{3}-l_{3})&\frac{% 1}{2}(j_{2}-l_{2}+j_{3}+l_{3})\\ l_{1}&\frac{1}{2}(j_{2}+l_{2}-j_{3}+l_{3})&\frac{1}{2}(-j_{2}+l_{2}+j_{3}+l_{3% })\end{Bmatrix},$
ⓘ Symbols: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $l,l_{r}$ : non-negative integers or non-negative integers plus one half. Referenced by: §34.5(ii) Permalink: http://dlmf.nist.gov/34.5.E9 Encodings: TeX, pMML, png See also: Annotations for §34.5(ii), §34.5 and Ch.34
34.5.10	$\displaystyle\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}$	$\displaystyle=\begin{Bmatrix}\frac{1}{2}(j_{2}+l_{2}+j_{3}-l_{3})&\frac{1}{2}(% j_{1}-l_{1}+j_{3}+l_{3})&\frac{1}{2}(j_{1}+l_{1}+j_{2}-l_{2})\\ \frac{1}{2}(j_{2}+l_{2}-j_{3}+l_{3})&\frac{1}{2}(-j_{1}+l_{1}+j_{3}+l_{3})&% \frac{1}{2}(j_{1}+l_{1}-j_{2}+l_{2})\end{Bmatrix}.$
ⓘ Symbols: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $l,l_{r}$ : non-negative integers or non-negative integers plus one half. Referenced by: §34.5(ii) Permalink: http://dlmf.nist.gov/34.5.E10 Encodings: TeX, pMML, png See also: Annotations for §34.5(ii), §34.5 and Ch.34

Equations (34.5.9) and (34.5.10) are called Regge symmetries. Additional symmetries are obtained by applying (34.5.8) to (34.5.9) and (34.5.10). See Srinivasa Rao and Rajeswari (1993, pp. 102–103) and references given there.

§34.5(iii) Recursion Relations

ⓘ

Keywords:: recursion relations, $\mathit{6j}$ symbols
Permalink:: http://dlmf.nist.gov/34.5.iii
See also:: Annotations for §34.5 and Ch.34

In the following equation it is assumed that the triangle conditions are satisfied.

34.5.11		${(2j_{1}+1)\left((J_{3}+J_{2}-J_{1})(L_{3}+L_{2}-J_{1})-2(J_{3}L_{3}+J_{2}L_{2% }-J_{1}L_{1})\right)\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}}\\ =j_{1}E(j_{1}+1)\begin{Bmatrix}j_{1}+1&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}+(j_{1}+1)E(j_{1})\begin{Bmatrix}j_{1}-1&j_{2}&j% _{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix},$
ⓘ Symbols: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $l,l_{r}$ : non-negative integers or non-negative integers plus one half., $J_{r}$ , $L_{r}$ and $E(j)$ Permalink: http://dlmf.nist.gov/34.5.E11 Encodings: TeX, pMML, png See also: Annotations for §34.5(iii), §34.5 and Ch.34

where

34.5.12	$\displaystyle J_{r}$	$\displaystyle=j_{r}(j_{r}+1),$
34.5.12	$\displaystyle L_{r}$	$\displaystyle=l_{r}(l_{r}+1),$
ⓘ Defines: $J_{r}$ (locally) and $L_{r}$ (locally) Symbols: $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $l,l_{r}$ : non-negative integers or non-negative integers plus one half. and $r$ : nonnegative integer Permalink: http://dlmf.nist.gov/34.5.E12 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §34.5(iii), §34.5 and Ch.34

34.5.13		$E(j)=\left((j^{2}-(j_{2}-j_{3})^{2})((j_{2}+j_{3}+1)^{2}-j^{2})(j^{2}-(l_{2}-l% _{3})^{2})((l_{2}+l_{3}+1)^{2}-j^{2})\right)^{\frac{1}{2}}.$
ⓘ Defines: $E(j)$ (locally) Symbols: $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $l,l_{r}$ : non-negative integers or non-negative integers plus one half. Permalink: http://dlmf.nist.gov/34.5.E13 Encodings: TeX, pMML, png See also: Annotations for §34.5(iii), §34.5 and Ch.34

For further recursion relations see Varshalovich et al. (1988, §9.6) and Edmonds (1974, pp. 98–99).

§34.5(iv) Orthogonality

ⓘ

Keywords:: orthogonality, $\mathit{6j}$ symbols
Notes:: See Edmonds (1974, p. 96).
Permalink:: http://dlmf.nist.gov/34.5.iv
See also:: Annotations for §34.5 and Ch.34

34.5.14		$\sum_{j_{3}}(2j_{3}+1)(2l_{3}+1)\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l^{\prime}_{3}\end{Bmatrix}=\delta_{l_{3},l^{\prime}_{3}}.$
ⓘ Symbols: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $l,l_{r}$ : non-negative integers or non-negative integers plus one half. Referenced by: §18.38(iii) Permalink: http://dlmf.nist.gov/34.5.E14 Encodings: TeX, pMML, png See also: Annotations for §34.5(iv), §34.5 and Ch.34

§34.5(v) Generating Functions

ⓘ

Keywords:: generating functions, $\mathit{6j}$ symbols
Permalink:: http://dlmf.nist.gov/34.5.v
See also:: Annotations for §34.5 and Ch.34

For generating functions for the $\mathit{6j}$ symbol see Biedenharn and van Dam (1965, p. 255, eq. (4.18)).

§34.5(vi) Sums

ⓘ

Keywords:: addition theorem, alternative, definition, $\mathit{6j}$ symbols, sum rules, sums
Notes:: See de-Shalit and Talmi (1963, pp. 517–518), Varshalovich et al. (1988, §9.8), Edmonds (1974, pp. 95–97), Dunlap and Judd (1975).
Permalink:: http://dlmf.nist.gov/34.5.vi
See also:: Annotations for §34.5 and Ch.34

34.5.15		$\sum_{j}(-1)^{j+j^{\prime}+j^{\prime\prime}}(2j+1)\begin{Bmatrix}j_{1}&j_{2}&j% \\ j_{3}&j_{4}&j^{\prime}\end{Bmatrix}\begin{Bmatrix}j_{1}&j_{2}&j\\ j_{4}&j_{3}&j^{\prime\prime}\end{Bmatrix}=\begin{Bmatrix}j_{1}&j_{4}&j^{\prime% }\\ j_{2}&j_{3}&j^{\prime\prime}\end{Bmatrix},$
ⓘ Symbols: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol and $j,j_{r}$ : non-negative integers or non-negative integers plus one half. Referenced by: §34.5(vi), §34.9 Permalink: http://dlmf.nist.gov/34.5.E15 Encodings: TeX, pMML, png See also: Annotations for §34.5(vi), §34.5 and Ch.34

34.5.16		$(-1)^{j_{1}+j_{2}+j_{3}+j_{1}^{\prime}+j_{2}^{\prime}+l_{1}+l_{2}}\begin{% Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}\begin{Bmatrix}j_{1}^{\prime}&j_{2}^{\prime}&j_{% 3}\\ l_{1}&l_{2}&l_{3}^{\prime}\end{Bmatrix}=\sum_{j}(-1)^{l_{3}+l_{3}^{\prime}+j}(% 2j+1)\begin{Bmatrix}j_{1}&j_{1}^{\prime}&j\\ j_{2}^{\prime}&j_{2}&j_{3}\end{Bmatrix}\begin{Bmatrix}l_{3}&l_{3}^{\prime}&j\\ j_{1}^{\prime}&j_{1}&l_{2}\end{Bmatrix}\begin{Bmatrix}l_{3}&l_{3}^{\prime}&j\\ j_{2}^{\prime}&j_{2}&l_{1}\end{Bmatrix}.$
ⓘ Symbols: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $l,l_{r}$ : non-negative integers or non-negative integers plus one half. Referenced by: §34.5(vi) Permalink: http://dlmf.nist.gov/34.5.E16 Encodings: TeX, pMML, png See also: Annotations for §34.5(vi), §34.5 and Ch.34

Equations (34.5.15) and (34.5.16) are the sum rules. They constitute addition theorems for the $\mathit{6j}$ symbol.

34.5.17	$\displaystyle\sum_{j}(2j+1)\begin{Bmatrix}j_{1}&j_{2}&j\\ j_{1}&j_{2}&j^{\prime}\end{Bmatrix}$	$\displaystyle=(-1)^{2(j_{1}+j_{2})},$
ⓘ Symbols: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol and $j,j_{r}$ : non-negative integers or non-negative integers plus one half. Permalink: http://dlmf.nist.gov/34.5.E17 Encodings: TeX, pMML, png See also: Annotations for §34.5(vi), §34.5 and Ch.34
34.5.18	$\displaystyle\sum_{j}(-1)^{j_{1}+j_{2}+j}(2j+1)\begin{Bmatrix}j_{1}&j_{2}&j\\ j_{2}&j_{1}&j^{\prime}\end{Bmatrix}$	$\displaystyle=\sqrt{(2j_{1}+1)(2j_{2}+1)}\,\delta_{j^{\prime},0},$
ⓘ Symbols: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol and $j,j_{r}$ : non-negative integers or non-negative integers plus one half. Permalink: http://dlmf.nist.gov/34.5.E18 Encodings: TeX, pMML, png See also: Annotations for §34.5(vi), §34.5 and Ch.34
34.5.19	$\displaystyle\sum_{l}\begin{Bmatrix}j_{1}&j_{2}&l\\ j_{2}&j_{1}&j\end{Bmatrix}$	$\displaystyle=0,$
$2\mu-j$ odd, $\mu=\min(j_{1},j_{2})$ ,
ⓘ Symbols: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $l,l_{r}$ : non-negative integers or non-negative integers plus one half. Permalink: http://dlmf.nist.gov/34.5.E19 Encodings: TeX, pMML, png See also: Annotations for §34.5(vi), §34.5 and Ch.34
34.5.20	$\displaystyle\sum_{l}(-1)^{l+j}\begin{Bmatrix}j_{1}&j_{2}&l\\ j_{1}&j_{2}&j\end{Bmatrix}$	$\displaystyle=\frac{(-1)^{2\mu}}{2j+1},$
$\mu=\min(j_{1},j_{2})$ ,
ⓘ Symbols: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $l,l_{r}$ : non-negative integers or non-negative integers plus one half. Permalink: http://dlmf.nist.gov/34.5.E20 Encodings: TeX, pMML, png See also: Annotations for §34.5(vi), §34.5 and Ch.34

34.5.21	$\displaystyle\sum_{l}(-1)^{l+j+j_{1}+j_{2}}\begin{Bmatrix}j_{1}&j_{2}&l\\ j_{2}&j_{1}&j\end{Bmatrix}$	$\displaystyle=\frac{1}{2j+1}\left(\frac{(2j_{1}-j)!(2j_{2}+j+1)!}{(2j_{2}-j)!(% 2j_{1}+j+1)!}\right)^{\frac{1}{2}},$
$j_{2}\leq j_{1}$ ,
ⓘ Symbols: $!$ : factorial (as in $n!$ ), $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $l,l_{r}$ : non-negative integers or non-negative integers plus one half. Permalink: http://dlmf.nist.gov/34.5.E21 Encodings: TeX, pMML, png See also: Annotations for §34.5(vi), §34.5 and Ch.34
34.5.22	$\displaystyle\sum_{l}(-1)^{l+j+j_{1}+j_{2}}\frac{1}{l(l+1)}\begin{Bmatrix}j_{1% }&j_{2}&l\\ j_{2}&j_{1}&j\end{Bmatrix}$	$\displaystyle=\frac{1}{j_{1}(j_{1}+1)-j_{2}(j_{2}+1)}\left(\frac{(2j_{1}-j)!(2% j_{2}+j+1)!}{(2j_{2}-j)!(2j_{1}+j+1)!}\right)^{\frac{1}{2}},$
$j_{2}<j_{1}$ .
ⓘ Symbols: $!$ : factorial (as in $n!$ ), $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $l,l_{r}$ : non-negative integers or non-negative integers plus one half. Permalink: http://dlmf.nist.gov/34.5.E22 Encodings: TeX, pMML, png See also: Annotations for §34.5(vi), §34.5 and Ch.34

34.5.23		$\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}=\sum_{m^{\prime}_{1}m^{\prime}_{2}m^{\prime}_{3% }}(-1)^{l_{1}+l_{2}+l_{3}+m^{\prime}_{1}+m^{\prime}_{2}+m^{\prime}_{3}}\begin{% pmatrix}j_{1}&l_{2}&l_{3}\\ m_{1}&m^{\prime}_{2}&-m^{\prime}_{3}\end{pmatrix}\begin{pmatrix}l_{1}&j_{2}&l_% {3}\\ -m^{\prime}_{1}&m_{2}&m^{\prime}_{3}\end{pmatrix}\begin{pmatrix}l_{1}&l_{2}&j_% {3}\\ m^{\prime}_{1}&-m^{\prime}_{2}&m_{3}\end{pmatrix}.$
ⓘ Symbols: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $l,l_{r}$ : non-negative integers or non-negative integers plus one half. Referenced by: §34.5(vi) Permalink: http://dlmf.nist.gov/34.5.E23 Encodings: TeX, pMML, png See also: Annotations for §34.5(vi), §34.5 and Ch.34