DLMF:14.9.E7 (Q4798)

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DLMF:14.9.E7
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    Statements

    test
    sin ⁡ ( ( ν - μ ) ⁢ π ) Γ ⁡ ( ν + μ + 1 ) ⁢ 𝖯 ν μ ⁡ ( x ) = sin ⁡ ( ν ⁢ π ) Γ ⁡ ( ν - μ + 1 ) ⁢ 𝖯 ν - μ ⁡ ( x ) - sin ⁡ ( μ ⁢ π ) Γ ⁡ ( ν - μ + 1 ) ⁢ 𝖯 ν - μ ⁡ ( - x ) , 𝜈 𝜇 𝜋 Euler-Gamma 𝜈 𝜇 1 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 𝜈 𝜋 Euler-Gamma 𝜈 𝜇 1 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 𝜇 𝜋 Euler-Gamma 𝜈 𝜇 1 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\frac{\sin\left((\nu-\mu)\pi\right)}{\Gamma\left(% \nu+\mu+1\right)}\mathsf{P}^{\mu}_{\nu}\left(x\right)=\frac{\sin\left(\nu\pi% \right)}{\Gamma\left(\nu-\mu+1\right)}\mathsf{P}^{-\mu}_{\nu}\left(x\right)-% \frac{\sin\left(\mu\pi\right)}{\Gamma\left(\nu-\mu+1\right)}\mathsf{P}^{-\mu}_% {\nu}\left(-x\right),}}
    0 references
    DLMF id
    DLMF:14.9.E7
    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
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    Ferrers function of the first kind
    test
    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.m2aedec
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    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
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    sine function
    test
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2acdec
    0 references
     
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