DLMF:14.18.E9 (Q4910): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Ferrers function of the first kind / rank
 
Normal rank
Property / Symbols used: Ferrers function of the first kind / qualifier
 
test:

𝖯 ν μ ⁡ ( x ) Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}% \right)}}

\FerrersP[\NVar{\mu}]{\NVar{\nu}}@{\NVar{x}}
Property / Symbols used: Ferrers function of the first kind / qualifier
 
xml-id: C14.S3.E1.m2acdec

Revision as of 12:28, 2 January 2020

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DLMF:14.18.E9
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    Statements

    test
    𝖯 ν - μ ⁡ ( x ) = sin ⁡ ( ν ⁢ π ) π ⁢ ∑ n = 0 ∞ ( - 1 ) n ⁢ 2 ⁢ n + 1 ( ν - n ) ⁢ ( ν + n + 1 ) ⁢ 𝖯 n - μ ⁡ ( x ) , Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 𝜈 𝜋 𝜋 superscript subscript 𝑛 0 superscript 1 𝑛 2 𝑛 1 𝜈 𝑛 𝜈 𝑛 1 Ferrers-Legendre-P-first-kind 𝜇 𝑛 𝑥 {\displaystyle{\displaystyle\mathsf{P}^{-\mu}_{\nu}\left(x\right)=\frac{\sin% \left(\nu\pi\right)}{\pi}\sum_{n=0}^{\infty}(-1)^{n}\frac{2n+1}{(\nu-n)(\nu+n+% 1)}\mathsf{P}^{-\mu}_{n}\left(x\right),}}
    0 references
    DLMF id
    DLMF:14.18.E9
    0 references
    constraint
    ν ∉ ℤ 𝜈 {\displaystyle{\displaystyle\nu\notin\mathbb{Z}}}
    0 references
    μ ≥ 0 𝜇 0 {\displaystyle{\displaystyle\mu\geq 0}}
    0 references
    ν ∉ ℤ 𝜈 ℤ {\displaystyle{\displaystyle\nu\notin\mathbb{Z}}}
    0 references
    Symbols used
    Ferrers function of the first kind
    test
    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.m2acdec
    0 references
     
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