DLMF:14.11.E2 (Q4818)

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DLMF:14.11.E2
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    ν 𝖰 ν μ ( x ) = - 1 2 π 2 𝖯 ν μ ( x ) + π sin ( μ π ) sin ( ν π ) sin ( ( ν + μ ) π ) 𝖰 ν μ ( x ) - 1 2 cot ( ( ν + μ ) π ) 𝖠 ν μ ( x ) + 1 2 csc ( ( ν + μ ) π ) 𝖠 ν μ ( - x ) , partial-derivative 𝜈 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 1 2 superscript 𝜋 2 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 𝜋 𝜇 𝜋 𝜈 𝜋 𝜈 𝜇 𝜋 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 1 2 𝜈 𝜇 𝜋 superscript subscript 𝖠 𝜈 𝜇 𝑥 1 2 𝜈 𝜇 𝜋 superscript subscript 𝖠 𝜈 𝜇 𝑥 {\displaystyle{\displaystyle\frac{\partial}{\partial\nu}\mathsf{Q}^{\mu}_{\nu}% \left(x\right)=-\tfrac{1}{2}\pi^{2}\mathsf{P}^{\mu}_{\nu}\left(x\right)+\frac{% \pi\sin\left(\mu\pi\right)}{\sin\left(\nu\pi\right)\sin\left((\nu+\mu)\pi% \right)}\mathsf{Q}^{\mu}_{\nu}\left(x\right)-\tfrac{1}{2}\cot\left((\nu+\mu)% \pi\right)\mathsf{A}_{\nu}^{\mu}(x)+\tfrac{1}{2}\csc\left((\nu+\mu)\pi\right)% \mathsf{A}_{\nu}^{\mu}(-x),}}
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    DLMF:14.11.E2
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    𝖯 ν μ ( x ) Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}% \right)}}
    C14.S3.E1.m2aadec
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