DLMF:14.9.E7 (Q4798)

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DLMF:14.9.E7
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    Statements

    Defining formula
    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),}}
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    DLMF id
    DLMF:14.9.E7
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