DLMF:14.12.E5 (Q4826)

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DLMF:14.12.E5
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    Statements

    test
    P ν - μ ⁡ ( x ) = ( x 2 - 1 ) - μ / 2 Γ ⁡ ( μ ) ⁢ ∫ 1 x P ν ⁡ ( t ) ⁢ ( x - t ) μ - 1 ⁢ d t , Legendre-P-first-kind 𝜇 𝜈 𝑥 superscript superscript 𝑥 2 1 𝜇 2 Euler-Gamma 𝜇 superscript subscript 1 𝑥 Legendre-spherical-polynomial 𝜈 𝑡 superscript 𝑥 𝑡 𝜇 1 𝑡 {\displaystyle{\displaystyle P^{-\mu}_{\nu}\left(x\right)=\frac{\left(x^{2}-1% \right)^{-\mu/2}}{\Gamma\left(\mu\right)}\int_{1}^{x}P_{\nu}\left(t\right)(x-t% )^{\mu-1}\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:14.12.E5
    0 references
    constraint
    ℜ ⁡ μ > 0 𝜇 0 {\displaystyle{\displaystyle\Re\mu>0}}
    0 references
    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2addec
    0 references
    Q11481
    test
    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.m1aadec
    0 references
     
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