DLMF:14.10.E4 (Q4813): 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
 
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)}}

\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 01:39, 2 January 2020

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DLMF:14.10.E4
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    Statements

    Defining formula
    ( 1 - x 2 ) ⁢ d 𝖯 ν μ ⁡ ( x ) d x = ( μ - ν - 1 ) ⁢ 𝖯 ν + 1 μ ⁡ ( x ) + ( ν + 1 ) ⁢ x ⁢ 𝖯 ν μ ⁡ ( x ) , 1 superscript 𝑥 2 derivative Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 𝑥 𝜇 𝜈 1 Ferrers-Legendre-P-first-kind 𝜇 𝜈 1 𝑥 𝜈 1 𝑥 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\left(1-x^{2}\right)\frac{\mathrm{d}\mathsf{P}^{% \mu}_{\nu}\left(x\right)}{\mathrm{d}x}={(\mu-\nu-1)\mathsf{P}^{\mu}_{\nu+1}% \left(x\right)+(\nu+1)x\mathsf{P}^{\mu}_{\nu}\left(x\right)},}}
    0 references
    DLMF id
    DLMF:14.10.E4
    0 references
    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.m2acdec
    0 references
     
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