DLMF:14.12.E2 (Q4823)

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DLMF:14.12.E2
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    Statements

    Defining formula
    𝖯 ν - μ ⁡ ( x ) = ( 1 - x 2 ) - μ / 2 Γ ⁡ ( μ ) ⁢ ∫ x 1 𝖯 ν ⁡ ( t ) ⁢ ( t - x ) μ - 1 ⁢ d t , Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 superscript 1 superscript 𝑥 2 𝜇 2 Euler-Gamma 𝜇 superscript subscript 𝑥 1 shorthand-Ferrers-Legendre-P-first-kind 𝜈 𝑡 superscript 𝑡 𝑥 𝜇 1 𝑡 {\displaystyle{\displaystyle\mathsf{P}^{-\mu}_{\nu}\left(x\right)=\frac{\left(% 1-x^{2}\right)^{-\mu/2}}{\Gamma\left(\mu\right)}\int_{x}^{1}\mathsf{P}_{\nu}% \left(t\right)(t-x)^{\mu-1}\mathrm{d}t,}}
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    DLMF id
    DLMF:14.12.E2
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    constraint
    ℜ ⁡ μ > 0 𝜇 0 {\displaystyle{\displaystyle\Re\mu>0}}
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    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aadec
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    Ferrers function of the first kind
    Defining formula
    𝖯 ν μ ⁡ ( x ) Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}% \right)}}
    xml-id
    C14.S3.E1.m2aadec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aadec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aadec
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