DLMF:14.17.E3 (Q4884)

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DLMF:14.17.E3
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    Statements

    Defining formula
    ∫ x ⁢ 𝖯 ν μ ⁡ ( x ) ⁢ 𝖰 ν μ ⁡ ( x ) ⁢ d x = 1 2 ⁢ ν ⁢ ( ν + 1 ) ⁢ ( ( μ 2 - ( ν + 1 ) ⁢ ( ν + x 2 ) ) ⁢ 𝖯 ν μ ⁡ ( x ) ⁢ 𝖰 ν μ ⁡ ( x ) + ( ν + 1 ) ⁢ ( ν - μ + 1 ) ⁢ x ⁢ ( 𝖯 ν μ ⁡ ( x ) ⁢ 𝖰 ν + 1 μ ⁡ ( x ) + 𝖯 ν + 1 μ ⁡ ( x ) ⁢ 𝖰 ν μ ⁡ ( x ) ) - ( ν - μ + 1 ) 2 ⁢ 𝖯 ν + 1 μ ⁡ ( x ) ⁢ 𝖰 ν + 1 μ ⁡ ( x ) ) , 𝑥 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 𝑥 1 2 𝜈 𝜈 1 superscript 𝜇 2 𝜈 1 𝜈 superscript 𝑥 2 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 𝜈 1 𝜈 𝜇 1 𝑥 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 1 𝑥 Ferrers-Legendre-P-first-kind 𝜇 𝜈 1 𝑥 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 superscript 𝜈 𝜇 1 2 Ferrers-Legendre-P-first-kind 𝜇 𝜈 1 𝑥 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 1 𝑥 {\displaystyle{\displaystyle\int x\mathsf{P}^{\mu}_{\nu}\left(x\right)\mathsf{% Q}^{\mu}_{\nu}\left(x\right)\mathrm{d}x=\frac{1}{2\nu(\nu+1)}\left((\mu^{2}-(% \nu+1)(\nu+x^{2}))\mathsf{P}^{\mu}_{\nu}\left(x\right)\mathsf{Q}^{\mu}_{\nu}% \left(x\right)+(\nu+1)(\nu-\mu+1)x(\mathsf{P}^{\mu}_{\nu}\left(x\right)\mathsf% {Q}^{\mu}_{\nu+1}\left(x\right)+\mathsf{P}^{\mu}_{\nu+1}\left(x\right)\mathsf{% Q}^{\mu}_{\nu}\left(x\right))-(\nu-\mu+1)^{2}\mathsf{P}^{\mu}_{\nu+1}\left(x% \right)\mathsf{Q}^{\mu}_{\nu+1}\left(x\right)\right),}}
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    DLMF id
    DLMF:14.17.E3
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    constraint
    ν ≠ 0 , - 1 𝜈 0 1 {\displaystyle{\displaystyle\nu\neq 0,-1}}
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