DLMF:14.17.E3 (Q4884)

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DLMF:14.17.E3
No description defined

    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),}}
    0 references
    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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    Symbols used
    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.m2abdec
    0 references
    Ferrers function of the second kind
    Defining formula
    𝖰 ν μ ⁡ ( x ) Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}% \right)}}
    xml-id
    C14.S3.E2.m2adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abdec
    0 references
    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1bdec
    0 references
    Q11591
    Defining formula
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C14.S1.XMD7.m1bdec
    0 references
    Q11590
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1bdec
    0 references
     
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