DLMF:14.28.E1 (Q4960)

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DLMF:14.28.E1
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    Statements

    Defining formula
    P ν ⁡ ( z 1 ⁢ z 2 - ( z 1 2 - 1 ) 1 / 2 ⁢ ( z 2 2 - 1 ) 1 / 2 ⁢ cos ⁡ ϕ ) = P ν ⁡ ( z 1 ) ⁢ P ν ⁡ ( z 2 ) + 2 ⁢ ∑ m = 1 ∞ ( - 1 ) m ⁢ Γ ⁡ ( ν - m + 1 ) Γ ⁡ ( ν + m + 1 ) ⁢ P ν m ⁡ ( z 1 ) ⁢ P ν m ⁡ ( z 2 ) ⁢ cos ⁡ ( m ⁢ ϕ ) , shorthand-Legendre-P-first-kind 𝜈 subscript 𝑧 1 subscript 𝑧 2 superscript superscript subscript 𝑧 1 2 1 1 2 superscript superscript subscript 𝑧 2 2 1 1 2 italic-ϕ shorthand-Legendre-P-first-kind 𝜈 subscript 𝑧 1 shorthand-Legendre-P-first-kind 𝜈 subscript 𝑧 2 2 superscript subscript 𝑚 1 superscript 1 𝑚 Euler-Gamma 𝜈 𝑚 1 Euler-Gamma 𝜈 𝑚 1 Legendre-P-first-kind 𝑚 𝜈 subscript 𝑧 1 Legendre-P-first-kind 𝑚 𝜈 subscript 𝑧 2 𝑚 italic-ϕ {\displaystyle{\displaystyle P_{\nu}\left(z_{1}z_{2}-\left(z_{1}^{2}-1\right)^% {1/2}\left(z_{2}^{2}-1\right)^{1/2}\cos\phi\right)=P_{\nu}\left(z_{1}\right)P_% {\nu}\left(z_{2}\right)+2\sum_{m=1}^{\infty}(-1)^{m}\frac{\Gamma\left(\nu-m+1% \right)}{\Gamma\left(\nu+m+1\right)}\*P^{m}_{\nu}\left(z_{1}\right)P^{m}_{\nu}% (z_{2})\cos\left(m\phi\right),}}
    0 references
    DLMF id
    DLMF:14.28.E1
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
    0 references
    Q11481
    Defining formula
    P ν μ ⁡ ( z ) Legendre-P-first-kind 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C14.S21.SS1.p1.m1adec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
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    Q11447
    Defining formula
    P ν ⁡ ( z ) = P ν 0 ⁡ ( z ) shorthand-Legendre-P-first-kind 𝜈 𝑧 Legendre-P-first-kind 0 𝜈 𝑧 {\displaystyle{\displaystyle P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}% \left(z\right)}}
    xml-id
    C14.S2.SS2.p2.m3adec
    0 references
    Q11638
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C14.S1.XMD4.m1dec
    0 references
    Q11593
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C14.S1.XMD5.m1dec
    0 references
    Q11590
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1dec
    0 references
     
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