DLMF:34.2.E6 (Q9715)

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DLMF:34.2.E6
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    Statements

    Defining formula
    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)},}}
    0 references
    DLMF id
    DLMF:34.2.E6
    0 references
    Symbols used
    Q11125
    Defining formula
    F q p ⁡ ( a 1 , … , a p ; b 1 , … , b q ; z ) Gauss-hypergeometric-pFq 𝑝 𝑞 subscript 𝑎 1 … subscript 𝑎 𝑝 subscript 𝑏 1 … subscript 𝑏 𝑞 𝑧 {\displaystyle{\displaystyle{{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots% ,a_{p}};\NVar{b_{1},\dots,b_{q}};\NVar{z}\right)}}
    xml-id
    C16.S2.m1adec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5abdec
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    $$\mathit{3j}$$ symbol
    Defining formula
    3 ⁢ j ⁢ j 1 ⁢ j 2 ⁢ j 3 ⁢ m 1 ⁢ missing threej subscript 𝑗 1 subscript 𝑗 2 subscript 𝑗 3 subscript 𝑚 1 missing {\displaystyle{\displaystyle\mathit{3j}{\NVar{j_{1}}}{\NVar{j_{2}}}{\NVar{j_{3% }}}{\NVar{m_{1}}}{\NVar{missing}\)\@add@PDF@RDFa@triples\end{document}}}}
    xml-id
    C34.S2.E4.m2aadec
    0 references
    Q12709
    Defining formula
    j , j r 𝑗 subscript 𝑗 𝑟 {\displaystyle{\displaystyle j,j_{r}}}
    xml-id
    C34.S1.XMD1.m1edec
    0 references
    product (locally)
    Defining formula
    Δ ⁢ ( j 1 ⁢ j 2 ⁢ j 3 ) Δ subscript 𝑗 1 subscript 𝑗 2 subscript 𝑗 3 {\displaystyle{\displaystyle\Delta(j_{1}j_{2}j_{3})}}
    xml-id
    C34.S2.XMD4.m1bdec
    0 references
     
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