DLMF:34.2.E6 (Q9715)

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DLMF:34.2.E6
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    3 j j 1 j 2 j 3 m 1 m 2 m 3 = ( - 1 ) j 2 - m 1 + m 3 ( j 1 + j 2 + m 3 ) ! ( j 2 + j 3 - m 1 ) ! Δ ( j 1 j 2 j 3 ) ( j 1 + j 2 + j 3 + 1 ) ! ( ( j 1 + m 1 ) ! ( j 3 - m 3 ) ! ( j 1 - m 1 ) ! ( j 2 + m 2 ) ! ( j 2 - m 2 ) ! ( j 3 + m 3 ) ! ) 1 2 F 2 3 ( - j 1 - j 2 - j 3 - 1 , - j 1 + m 1 , - j 3 - m 3 ; - j 1 - j 2 - m 3 , - j 2 - j 3 + m 1 ; 1 ) , threej subscript 𝑗 1 subscript 𝑗 2 subscript 𝑗 3 subscript 𝑚 1 subscript 𝑚 2 subscript 𝑚 3 superscript 1 subscript 𝑗 2 subscript 𝑚 1 subscript 𝑚 3 subscript 𝑗 1 subscript 𝑗 2 subscript 𝑚 3 subscript 𝑗 2 subscript 𝑗 3 subscript 𝑚 1 Δ subscript 𝑗 1 subscript 𝑗 2 subscript 𝑗 3 subscript 𝑗 1 subscript 𝑗 2 subscript 𝑗 3 1 superscript subscript 𝑗 1 subscript 𝑚 1 subscript 𝑗 3 subscript 𝑚 3 subscript 𝑗 1 subscript 𝑚 1 subscript 𝑗 2 subscript 𝑚 2 subscript 𝑗 2 subscript 𝑚 2 subscript 𝑗 3 subscript 𝑚 3 1 2 Gauss-hypergeometric-pFq 3 2 subscript 𝑗 1 subscript 𝑗 2 subscript 𝑗 3 1 subscript 𝑗 1 subscript 𝑚 1 subscript 𝑗 3 subscript 𝑚 3 subscript 𝑗 1 subscript 𝑗 2 subscript 𝑚 3 subscript 𝑗 2 subscript 𝑗 3 subscript 𝑚 1 1 {\displaystyle{\displaystyle\mathit{3j}{j_{1}}{j_{2}}{j_{3}}{m_{1}}{m_{2}}{m_{% 3}}={(-1)^{j_{2}-m_{1}+m_{3}}}\frac{(j_{1}+j_{2}+m_{3})!(j_{2}+j_{3}-m_{1})!}{% \Delta(j_{1}j_{2}j_{3})(j_{1}+j_{2}+j_{3}+1)!}\left(\frac{(j_{1}+m_{1})!(j_{3}% -m_{3})!}{(j_{1}-m_{1})!(j_{2}+m_{2})!(j_{2}-m_{2})!(j_{3}+m_{3})!}\right)^{% \frac{1}{2}}\*{{{}_{3}F_{2}}\left(-j_{1}-j_{2}-j_{3}-1,-j_{1}+m_{1},-j_{3}-m_{% 3};-j_{1}-j_{2}-m_{3},-j_{2}-j_{3}+m_{1};1\right)},}}
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    DLMF:34.2.E6
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