DLMF:14.17.E8 (Q4889)

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DLMF:14.17.E8
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    Statements

    Defining formula
    ∫ - 1 1 𝖯 n l ⁡ ( x ) ⁢ 𝖯 n m ⁡ ( x ) 1 - x 2 ⁢ d x = ( n + m ) ! ( n - m ) ! ⁢ m ⁢ δ l , m , superscript subscript 1 1 Ferrers-Legendre-P-first-kind 𝑙 𝑛 𝑥 Ferrers-Legendre-P-first-kind 𝑚 𝑛 𝑥 1 superscript 𝑥 2 𝑥 𝑛 𝑚 𝑛 𝑚 𝑚 Kronecker 𝑙 𝑚 {\displaystyle{\displaystyle\int_{-1}^{1}\frac{\mathsf{P}^{l}_{n}\left(x\right% )\mathsf{P}^{m}_{n}\left(x\right)}{1-x^{2}}\mathrm{d}x=\frac{(n+m)!}{(n-m)!m}% \delta_{l,m},}}
    0 references
    DLMF id
    DLMF:14.17.E8
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    constraint
    m > 0 𝑚 0 {\displaystyle{\displaystyle m>0}}
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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.m2agdec
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    Q10768
    Defining formula
    δ j , k Kronecker 𝑗 𝑘 {\displaystyle{\displaystyle\delta_{\NVar{j},\NVar{k}}}}
    xml-id
    introduction.Sx4.p1.t1.r4.m5abdec
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