Symbol 34.8 Approximations for Large Parameters

§34.7 Basic Properties: Symbol

Permalink:: http://dlmf.nist.gov/34.7

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

§34.7(i) Special Case

Notes:: See Edmonds (1974, pp. 105–106), de-Shalit and Talmi (1963, p. 517).
Keywords:: symbols
Permalink:: http://dlmf.nist.gov/34.7.i

34.7.1 $\begin{Bmatrix}j_{{11}}&j_{{12}}&j_{{13}}\\ j_{{21}}&j_{{22}}&j_{{13}}\\ j_{{31}}&j_{{31}}&0\end{Bmatrix}=\frac{(-1)^{{j_{{12}}+j_{{21}}+j_{{13}}+j_{{3% 1}}}}}{((2j_{{13}}+1)(2j_{{31}}+1))^{{\frac{1}{2}}}}\begin{Bmatrix}j_{{11}}&j_% {{12}}&j_{{13}}\\ j_{{22}}&j_{{21}}&j_{{31}}\end{Bmatrix}.$

Symbols:: $\begin{Bmatrix}j_{{11}}&j_{{12}}&j_{{13}}\\ j_{{21}}&j_{{22}}&j_{{23}}\\ j_{{31}}&j_{{32}}&j_{{33}}\end{Bmatrix}$ : symbol, $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : symbol and $j,j_{r}$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.7.E1
Encodings:: TeX, pMML, png

§34.7(ii) Symmetry

Keywords:: symbols
Permalink:: http://dlmf.nist.gov/34.7.ii

The symbol has symmetry properties with respect to permutation of columns, permutation of rows, and transposition of rows and columns; these relate 72 independent symbols. Even (cyclic) permutations of either columns or rows, as well as transpositions, leave the symbol unchanged. Odd permutations of columns or rows introduce a phase factor $(-1)^{R}$ , where is the sum of all arguments of the symbol.

For further symmetry properties of the symbol see Edmonds (1974, pp. 102–103) and Varshalovich et al. (1988, §10.4.1).

§34.7(iii) Recursion Relations

Keywords:: symbols
Permalink:: http://dlmf.nist.gov/34.7.iii

For recursion relations see Varshalovich et al. (1988, §10.5).

§34.7(iv) Orthogonality

Notes:: See Edmonds (1974, p. 103).
Keywords:: symbols
Permalink:: http://dlmf.nist.gov/34.7.iv

34.7.2 $\sum_{{j_{{12}}\,j_{{34}}}}(2j_{{12}}+1)(2j_{{34}}+1)(2j_{{13}}+1)(2j_{{24}}+1% )\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{12}}\\ j_{{3}}&j_{{4}}&j_{{34}}\\ j_{{13}}&j_{{24}}&j\end{Bmatrix}\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{12}}\\ j_{{3}}&j_{{4}}&j_{{34}}\\ j^{{\prime}}_{{13}}&j^{{\prime}}_{{24}}&j\end{Bmatrix}=\delta_{{j_{{13}},j^{{% \prime}}_{{13}}}}\delta_{{j_{{24}},j^{{\prime}}_{{24}}}}.$

Symbols:: $\delta_{{j,k}}$ : Kronecker delta, $\begin{Bmatrix}j_{{11}}&j_{{12}}&j_{{13}}\\ j_{{21}}&j_{{22}}&j_{{23}}\\ j_{{31}}&j_{{32}}&j_{{33}}\end{Bmatrix}$ : symbol and $j,j_{r}$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.7.E2
Encodings:: TeX, pMML, png

§34.7(v) Generating Functions

Keywords:: symbols
Permalink:: http://dlmf.nist.gov/34.7.v

For generating functions for the symbol see Biedenharn and van Dam (1965, p. 258, eq. (4.37)).

§34.7(vi) Sums

Notes:: See Edmonds (1974, pp. 103–104), de-Shalit and Talmi (1963, pp. 127, 518).
Keywords:: symbols
Permalink:: http://dlmf.nist.gov/34.7.vi

34.7.3 $\sum_{{j_{{13}}\,j_{{24}}}}(-1)^{{2j_{2}+j_{{24}}+j_{{23}}-j_{{34}}}}(2j_{{13}% }+1)(2j_{{24}}+1)\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{12}}\\ j_{{3}}&j_{{4}}&j_{{34}}\\ j_{{13}}&j_{{24}}&j\end{Bmatrix}\begin{Bmatrix}j_{{1}}&j_{{3}}&j_{{13}}\\ j_{{4}}&j_{{2}}&j_{{24}}\\ j_{{14}}&j_{{23}}&j\end{Bmatrix}=\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{12}}\\ j_{{4}}&j_{{3}}&j_{{34}}\\ j_{{14}}&j_{{23}}&j\end{Bmatrix}.$

Symbols:: $\begin{Bmatrix}j_{{11}}&j_{{12}}&j_{{13}}\\ j_{{21}}&j_{{22}}&j_{{23}}\\ j_{{31}}&j_{{32}}&j_{{33}}\end{Bmatrix}$ : symbol and $j,j_{r}$ : nonnegative integer
Referenced by:: §34.9
Permalink:: http://dlmf.nist.gov/34.7.E3
Encodings:: TeX, pMML, png

This equation is the sum rule. It constitutes an addition theorem for the symbol.

34.7.4 $\begin{pmatrix}j_{{13}}&j_{{23}}&j_{{33}}\\ m_{{13}}&m_{{23}}&m_{{33}}\end{pmatrix}\begin{Bmatrix}j_{{11}}&j_{{12}}&j_{{13% }}\\ j_{{21}}&j_{{22}}&j_{{23}}\\ j_{{31}}&j_{{32}}&j_{{33}}\end{Bmatrix}=\sum_{{m_{{r1}},m_{{r2}},r=1,2,3}}% \begin{pmatrix}j_{{11}}&j_{{12}}&j_{{13}}\\ m_{{11}}&m_{{12}}&m_{{13}}\end{pmatrix}\begin{pmatrix}j_{{21}}&j_{{22}}&j_{{23% }}\\ m_{{21}}&m_{{22}}&m_{{23}}\end{pmatrix}\*\begin{pmatrix}j_{{31}}&j_{{32}}&j_{{% 33}}\\ m_{{13}}&m_{{23}}&m_{{33}}\end{pmatrix}\begin{pmatrix}j_{{11}}&j_{{21}}&j_{{31% }}\\ m_{{11}}&m_{{21}}&m_{{31}}\end{pmatrix}\begin{pmatrix}j_{{12}}&j_{{22}}&j_{{32% }}\\ m_{{12}}&m_{{22}}&m_{{32}}\end{pmatrix}.$

Symbols:: $\begin{Bmatrix}j_{{11}}&j_{{12}}&j_{{13}}\\ j_{{21}}&j_{{22}}&j_{{23}}\\ j_{{31}}&j_{{32}}&j_{{33}}\end{Bmatrix}$ : symbol, $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$ : symbol, $j,j_{r}$ : nonnegative integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.7.E4
Encodings:: TeX, pMML, png

34.7.5 $\sum_{{j^{{\prime}}}}(2j^{{\prime}}+1)\begin{Bmatrix}j_{{11}}&j_{{12}}&j^{{% \prime}}\\ j_{{21}}&j_{{22}}&j_{{23}}\\ j_{{31}}&j_{{32}}&j_{{33}}\end{Bmatrix}\begin{Bmatrix}j_{{11}}&j_{{12}}&j^{{% \prime}}\\ j_{{23}}&j_{{33}}&j\end{Bmatrix}={(-1)^{{2j}}}\begin{Bmatrix}j_{{21}}&j_{{22}}% &j_{{23}}\\ j_{{12}}&j&j_{{32}}\end{Bmatrix}\begin{Bmatrix}j_{{31}}&j_{{32}}&j_{{33}}\\ j&j_{{11}}&j_{{21}}\end{Bmatrix}.$

Symbols:: $\begin{Bmatrix}j_{{11}}&j_{{12}}&j_{{13}}\\ j_{{21}}&j_{{22}}&j_{{23}}\\ j_{{31}}&j_{{32}}&j_{{33}}\end{Bmatrix}$ : symbol, $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : symbol and $j,j_{r}$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.7.E5
Encodings:: TeX, pMML, png