34 3j, 6j, 9j Symbols Properties 34.2 Definition:

\mathit{3j}

\mathit{6j}

Symbol

§34.3 Basic Properties: $\mathit{3j}$ Symbol

ⓘ

Referenced by:: Erratum (V1.1.6) for Chapters 10 Bessel Functions, 18 Orthogonal Polynomials, 34 3j, 6j, 9j Symbols
Permalink:: http://dlmf.nist.gov/34.3
See also:: Annotations for Ch.34

§34.3(i) Special Cases
§34.3(ii) Symmetry
§34.3(iii) Recursion Relations
§34.3(iv) Orthogonality
§34.3(v) Generating Functions
§34.3(vi) Sums
§34.3(vii) Relations to Legendre Polynomials and Spherical Harmonics

§34.3(i) Special Cases

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Keywords:: special cases, $\mathit{3j}$ symbols
Notes:: See Edmonds (1974, pp. 48–50) and de-Shalit and Talmi (1963, p. 519).
Permalink:: http://dlmf.nist.gov/34.3.i
See also:: Annotations for §34.3 and Ch.34

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

34.3.1	$\displaystyle\begin{pmatrix}j&j&0\\ m&-m&0\end{pmatrix}$	$\displaystyle=\frac{(-1)^{j-m}}{(2j+1)^{\frac{1}{2}}},$
ⓘ 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 and $j,j_{r}$ : non-negative integers or non-negative integers plus one half. Permalink: http://dlmf.nist.gov/34.3.E1 Encodings: TeX, pMML, png See also: Annotations for §34.3(i), §34.3 and Ch.34
34.3.2	$\displaystyle\begin{pmatrix}j&j&1\\ m&-m&0\end{pmatrix}$	$\displaystyle=(-1)^{j-m}\frac{2m}{\left(2j(2j+1)(2j+2)\right)^{\frac{1}{2}}},$
$j\geq\tfrac{1}{2}$ ,
ⓘ 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 and $j,j_{r}$ : non-negative integers or non-negative integers plus one half. Permalink: http://dlmf.nist.gov/34.3.E2 Encodings: TeX, pMML, png See also: Annotations for §34.3(i), §34.3 and Ch.34
34.3.3	$\displaystyle\begin{pmatrix}j&j&1\\ m&-m-1&1\end{pmatrix}$	$\displaystyle=(-1)^{j-m}\left(\frac{2(j-m)(j+m+1)}{2j(2j+1)(2j+2)}\right)^{% \frac{1}{2}},$
$j\geq\tfrac{1}{2}$ .
ⓘ 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 and $j,j_{r}$ : non-negative integers or non-negative integers plus one half. Permalink: http://dlmf.nist.gov/34.3.E3 Encodings: TeX, pMML, png See also: Annotations for §34.3(i), §34.3 and Ch.34

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		$J=j_{1}+j_{2}+j_{3}.$
ⓘ Defines: $J$ : sum (locally) Symbols: $j,j_{r}$ : non-negative integers or non-negative integers plus one half. Referenced by: §34.5(i) Permalink: http://dlmf.nist.gov/34.3.E4 Encodings: TeX, pMML, png See also: Annotations for §34.3(i), §34.3 and Ch.34

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{\left(\frac{1}{2}J\right)!}{\left(\frac{1}{2}J-j_{% 1}\right)!\left(\frac{1}{2}J-j_{2}\right)!\left(\frac{1}{2}J-j_{3}\right)!},&% \mbox{$J$ even}.\end{cases}$
ⓘ Symbols: $!$ : factorial (as in $n!$ ), $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $J$ : sum Permalink: http://dlmf.nist.gov/34.3.E5 Encodings: TeX, pMML, png See also: Annotations for §34.3(i), §34.3 and Ch.34

Lastly,

34.3.6		$\begin{pmatrix}j_{1}&j_{2}&j_{1}+j_{2}\\ m_{1}&m_{2}&-m_{1}-m_{2}\end{pmatrix}=(-1)^{j_{1}-j_{2}+m_{1}+m_{2}}\left(% \frac{(2j_{1})!(2j_{2})!(j_{1}+j_{2}+m_{1}+m_{2})!(j_{1}+j_{2}-m_{1}-m_{2})!}{% (2j_{1}+2j_{2}+1)!(j_{1}+m_{1})!(j_{1}-m_{1})!(j_{2}+m_{2})!(j_{2}-m_{2})!}% \right)^{\frac{1}{2}},$
ⓘ Symbols: $!$ : factorial (as in $n!$ ), $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol and $j,j_{r}$ : non-negative integers or non-negative integers plus one half. Permalink: http://dlmf.nist.gov/34.3.E6 Encodings: TeX, pMML, png See also: Annotations for §34.3(i), §34.3 and Ch.34

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: $!$ : factorial (as in $n!$ ), $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol and $j,j_{r}$ : non-negative integers or non-negative integers plus one half. Referenced by: §34.3(i), Erratum (V1.0.9) for 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 See also: Annotations for §34.3(i), §34.3 and Ch.34

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

§34.3(ii) Symmetry

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Keywords:: $\mathit{3j}$ symbols, Regge symmetries, symmetry, $\mathit{3j}$ symbols
Notes:: See Edmonds (1974, pp. 46–47).
Permalink:: http://dlmf.nist.gov/34.3.ii
See also:: Annotations for §34.3 and Ch.34

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

34.3.8	$\displaystyle\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$	$\displaystyle=\begin{pmatrix}j_{2}&j_{3}&j_{1}\\ m_{2}&m_{3}&m_{1}\end{pmatrix}=\begin{pmatrix}j_{3}&j_{1}&j_{2}\\ m_{3}&m_{1}&m_{2}\end{pmatrix},$
ⓘ 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 and $j,j_{r}$ : non-negative integers or non-negative integers plus one half. Referenced by: §34.3(ii) Permalink: http://dlmf.nist.gov/34.3.E8 Encodings: TeX, pMML, png See also: Annotations for §34.3(ii), §34.3 and Ch.34
34.3.9	$\displaystyle\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$	$\displaystyle=(-1)^{j_{1}+j_{2}+j_{3}}\begin{pmatrix}j_{2}&j_{1}&j_{3}\\ m_{2}&m_{1}&m_{3}\end{pmatrix}.$
ⓘ 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 and $j,j_{r}$ : non-negative integers or non-negative integers plus one half. Permalink: http://dlmf.nist.gov/34.3.E9 Encodings: TeX, pMML, png See also: Annotations for §34.3(ii), §34.3 and Ch.34

Next,

34.3.10	$\displaystyle\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$	$\displaystyle=(-1)^{j_{1}+j_{2}+j_{3}}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ -m_{1}&-m_{2}&-m_{3}\end{pmatrix},$
ⓘ 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 and $j,j_{r}$ : non-negative integers or non-negative integers plus one half. Referenced by: §34.3(ii) Permalink: http://dlmf.nist.gov/34.3.E10 Encodings: TeX, pMML, png See also: Annotations for §34.3(ii), §34.3 and Ch.34
34.3.11	$\displaystyle\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$	$\displaystyle=\begin{pmatrix}j_{1}&\frac{1}{2}(j_{2}+j_{3}+m_{1})&\frac{1}{2}(% j_{2}+j_{3}-m_{1})\\ j_{2}-j_{3}&\frac{1}{2}(j_{3}-j_{2}+m_{1})+m_{2}&\frac{1}{2}(j_{3}-j_{2}+m_{1}% )+m_{3}\end{pmatrix},$
ⓘ 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 and $j,j_{r}$ : non-negative integers or non-negative integers plus one half. Referenced by: §34.3(ii) Permalink: http://dlmf.nist.gov/34.3.E11 Encodings: TeX, pMML, png See also: Annotations for §34.3(ii), §34.3 and Ch.34
34.3.12	$\displaystyle\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$	$\displaystyle=\begin{pmatrix}\frac{1}{2}(j_{1}+j_{2}-m_{3})&\frac{1}{2}(j_{2}+% j_{3}-m_{1})&\frac{1}{2}(j_{1}+j_{3}-m_{2})\\ j_{3}-\frac{1}{2}(j_{1}+j_{2}+m_{3})&j_{1}-\frac{1}{2}(j_{2}+j_{3}+m_{1})&j_{2% }-\frac{1}{2}(j_{1}+j_{3}+m_{2})\end{pmatrix}.$
ⓘ 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 and $j,j_{r}$ : non-negative integers or non-negative integers plus one half. Referenced by: §34.3(ii) Permalink: http://dlmf.nist.gov/34.3.E12 Encodings: TeX, pMML, png See also: Annotations for §34.3(ii), §34.3 and Ch.34

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

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Keywords:: recursion relations, $\mathit{3j}$ symbols
Notes:: See Edmonds (1974, p. 48).
Permalink:: http://dlmf.nist.gov/34.3.iii
See also:: Annotations for §34.3 and Ch.34

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

34.3.13		$\left((j_{1}+j_{2}+j_{3}+1)(-j_{1}+j_{2}+j_{3})\right)^{\frac{1}{2}}\begin{% pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}=\left((j_{2}+m_{2})(j_{3}-m_{3})\right)^{\frac{% 1}{2}}\begin{pmatrix}j_{1}&j_{2}-\frac{1}{2}&j_{3}-\frac{1}{2}\\ m_{1}&m_{2}-\frac{1}{2}&m_{3}+\frac{1}{2}\end{pmatrix}-\left((j_{2}-m_{2})(j_{% 3}+m_{3})\right)^{\frac{1}{2}}\begin{pmatrix}j_{1}&j_{2}-\frac{1}{2}&j_{3}-% \frac{1}{2}\\ m_{1}&m_{2}+\frac{1}{2}&m_{3}-\frac{1}{2}\end{pmatrix},$
ⓘ 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 and $j,j_{r}$ : non-negative integers or non-negative integers plus one half. Permalink: http://dlmf.nist.gov/34.3.E13 Encodings: TeX, pMML, png See also: Annotations for §34.3(iii), §34.3 and Ch.34

34.3.14		$\left(j_{1}(j_{1}+1)-j_{2}(j_{2}+1)-j_{3}(j_{3}+1)-2m_{2}m_{3}\right)\begin{% pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}=\left((j_{2}-m_{2})(j_{2}+m_{2}+1)(j_{3}-m_{3}+% 1)(j_{3}+m_{3})\right)^{\frac{1}{2}}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}+1&m_{3}-1\end{pmatrix}+\left((j_{2}-m_{2}+1)(j_{2}+m_{2})(j_{3}-m_% {3})(j_{3}+m_{3}+1)\right)^{\frac{1}{2}}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}-1&m_{3}+1\end{pmatrix},$
ⓘ 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 and $j,j_{r}$ : non-negative integers or non-negative integers plus one half. Permalink: http://dlmf.nist.gov/34.3.E14 Encodings: TeX, pMML, png See also: Annotations for §34.3(iii), §34.3 and Ch.34

34.3.15		$(2j_{1}+1)\left((j_{2}(j_{2}+1)-j_{3}(j_{3}+1))m_{1}-j_{1}(j_{1}+1)(m_{3}-m_{2% })\right)\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}=(j_{1}+1)\left(j_{1}^{2}-(j_{2}-j_{3})^{2}% \right)^{\frac{1}{2}}\left((j_{2}+j_{3}+1)^{2}-j_{1}^{2}\right)^{\frac{1}{2}}% \left(j_{1}^{2}-m_{1}^{2}\right)^{\frac{1}{2}}\begin{pmatrix}j_{1}-1&j_{2}&j_{% 3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}+j_{1}\left((j_{1}+1)^{2}-(j_{2}-j_{3})^{2}% \right)^{\frac{1}{2}}\left((j_{2}+j_{3}+1)^{2}-(j_{1}+1)^{2}\right)^{\frac{1}{% 2}}\left((j_{1}+1)^{2}-m_{1}^{2}\right)^{\frac{1}{2}}\begin{pmatrix}j_{1}+1&j_% {2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}.$
ⓘ 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 and $j,j_{r}$ : non-negative integers or non-negative integers plus one half. Permalink: http://dlmf.nist.gov/34.3.E15 Encodings: TeX, pMML, png See also: Annotations for §34.3(iii), §34.3 and Ch.34

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

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Keywords:: $\mathit{9j}$ symbols, orthogonality, $\mathit{6j}$ symbols, summation convention, $\mathit{3j}$ symbols
Notes:: See Edmonds (1974, p. 47), de-Shalit and Talmi (1963, p. 515).
Referenced by:: §18.38(iii)
Permalink:: http://dlmf.nist.gov/34.3.iv
See also:: Annotations for §34.3 and Ch.34

34.3.16	$\displaystyle\sum_{m_{1}m_{2}}(2j_{3}+1)\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}\begin{pmatrix}j_{1}&j_{2}&j^{\prime}_{3}\\ m_{1}&m_{2}&m^{\prime}_{3}\end{pmatrix}$	$\displaystyle=\delta_{j_{3},j^{\prime}_{3}}\delta_{m_{3},m^{\prime}_{3}},$
ⓘ Symbols: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol and $j,j_{r}$ : non-negative integers or non-negative integers plus one half. Referenced by: §34.3(iv), §34.9 Permalink: http://dlmf.nist.gov/34.3.E16 Encodings: TeX, pMML, png See also: Annotations for §34.3(iv), §34.3 and Ch.34
34.3.17	$\displaystyle\sum_{j_{3}m_{3}}(2j_{3}+1)\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m^{\prime}_{1}&m^{\prime}_{2}&m_{3}\end{pmatrix}$	$\displaystyle=\delta_{m_{1},m^{\prime}_{1}}\delta_{m_{2},m^{\prime}_{2}},$
ⓘ Symbols: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol and $j,j_{r}$ : non-negative integers or non-negative integers plus one half. Permalink: http://dlmf.nist.gov/34.3.E17 Encodings: TeX, pMML, png See also: Annotations for §34.3(iv), §34.3 and Ch.34
34.3.18	$\displaystyle\sum_{m_{1}m_{2}m_{3}}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$	$\displaystyle=1.$
ⓘ 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 and $j,j_{r}$ : non-negative integers or non-negative integers plus one half. Referenced by: §34.3(iv) Permalink: http://dlmf.nist.gov/34.3.E18 Encodings: TeX, pMML, png See also: Annotations for §34.3(iv), §34.3 and Ch.34

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

ⓘ

Keywords:: generating functions, $\mathit{3j}$ symbols
Permalink:: http://dlmf.nist.gov/34.3.v
See also:: Annotations for §34.3 and Ch.34

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

§34.3(vi) Sums

ⓘ

Keywords:: sums, $\mathit{3j}$ symbols
Permalink:: http://dlmf.nist.gov/34.3.vi
See also:: Annotations for §34.3 and Ch.34

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

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

ⓘ

Keywords:: $\mathit{3j}$ symbol, Gaunt coefficient, Gaunt’s integral, Legendre functions, Legendre polynomials, relation to $\mathit{3j}$ symbols, relations to other functions, rotation matrices, spherical harmonics, $\mathit{3j}$ symbols
Notes:: See Thompson (1994, p. 288), Edmonds (1974, p. 63).
Referenced by:: Erratum (V1.0.19) for Notation
Permalink:: http://dlmf.nist.gov/34.3.vii
See also:: Annotations for §34.3 and Ch.34

For the polynomials $P_{l}$ see §18.3, and for the function $Y_{{l},{m}}$ see §14.30.

34.3.19		$P_{l_{1}}\left(\cos\theta\right)P_{l_{2}}\left(\cos\theta\right)=\sum_{l}(2l+1% )\begin{pmatrix}l_{1}&l_{2}&l\\ 0&0&0\end{pmatrix}^{2}P_{l}\left(\cos\theta\right),$
ⓘ Symbols: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\cos\NVar{z}$ : cosine function, $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol and $l,l_{r}$ : non-negative integers or non-negative integers plus one half. Referenced by: §14.18(iv), §14.30(iii), §34.3(vii) Permalink: http://dlmf.nist.gov/34.3.E19 Encodings: TeX, pMML, png See also: Annotations for §34.3(vii), §34.3 and Ch.34

34.3.20		$Y_{{l_{1}},{m_{1}}}\left(\theta,\phi\right)Y_{{l_{2}},{m_{2}}}\left(\theta,% \phi\right)=\sum_{l,m}\left(\frac{(2l_{1}+1)(2l_{2}+1)(2l+1)}{4\pi}\right)^{% \frac{1}{2}}\begin{pmatrix}l_{1}&l_{2}&l\\ m_{1}&m_{2}&m\end{pmatrix}\overline{Y_{{l},{m}}\left(\theta,\phi\right)}\begin% {pmatrix}l_{1}&l_{2}&l\\ 0&0&0\end{pmatrix},$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\overline{\NVar{z}}$ : complex conjugate, $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$ : spherical harmonic, $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol and $l,l_{r}$ : non-negative integers or non-negative integers plus one half. Permalink: http://dlmf.nist.gov/34.3.E20 Encodings: TeX, pMML, png Notational Change (effective with 1.0.19): As a notational clarification, the spherical harmonic in the sum over $l, m$ has been explicity marked up as a complex conjugate. See also: Annotations for §34.3(vii), §34.3 and Ch.34

34.3.21		$\int_{0}^{\pi}P_{l_{1}}\left(\cos\theta\right)P_{l_{2}}\left(\cos\theta\right)% P_{l_{3}}\left(\cos\theta\right)\sin\theta\,\mathrm{d}\theta=2\begin{pmatrix}l% _{1}&l_{2}&l_{3}\\ 0&0&0\end{pmatrix}^{2},$
ⓘ Symbols: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol and $l,l_{r}$ : non-negative integers or non-negative integers plus one half. Referenced by: §14.17(iv) Permalink: http://dlmf.nist.gov/34.3.E21 Encodings: TeX, pMML, png See also: Annotations for §34.3(vii), §34.3 and Ch.34

34.3.22		$\int_{0}^{2\pi}\!\int_{0}^{\pi}Y_{{l_{1}},{m_{1}}}\left(\theta,\phi\right)Y_{{% l_{2}},{m_{2}}}\left(\theta,\phi\right)Y_{{l_{3}},{m_{3}}}\left(\theta,\phi% \right)\sin\theta\,\mathrm{d}\theta\,\mathrm{d}\phi=\left(\frac{(2l_{1}+1)(2l_% {2}+1)(2l_{3}+1)}{4\pi}\right)^{\frac{1}{2}}\begin{pmatrix}l_{1}&l_{2}&l_{3}\\ 0&0&0\end{pmatrix}\begin{pmatrix}l_{1}&l_{2}&l_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$ : spherical harmonic, $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol and $l,l_{r}$ : non-negative integers or non-negative integers plus one half. Referenced by: §14.30(ii), §34.3(vii) Permalink: http://dlmf.nist.gov/34.3.E22 Encodings: TeX, pMML, png See also: Annotations for §34.3(vii), §34.3 and Ch.34

Equations (34.3.19)–(34.3.22) are particular cases of more general results that relate rotation matrices to $\mathit{3j}$ 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)).