DLMF:14.2.E5 (Q4683)

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DLMF:14.2.E5
No description defined

    Statements

    Defining formula
    𝖯 ν + 1 μ ⁡ ( x ) ⁢ 𝖰 ν μ ⁡ ( x ) - 𝖯 ν μ ⁡ ( x ) ⁢ 𝖰 ν + 1 μ ⁡ ( x ) = Γ ⁡ ( ν + μ + 1 ) Γ ⁡ ( ν - μ + 2 ) , Ferrers-Legendre-P-first-kind 𝜇 𝜈 1 𝑥 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 1 𝑥 Euler-Gamma 𝜈 𝜇 1 Euler-Gamma 𝜈 𝜇 2 {\displaystyle{\displaystyle\mathsf{P}^{\mu}_{\nu+1}\left(x\right)\mathsf{Q}^{% \mu}_{\nu}\left(x\right)-\mathsf{P}^{\mu}_{\nu}\left(x\right)\mathsf{Q}^{\mu}_% {\nu+1}\left(x\right)=\frac{\Gamma\left(\nu+\mu+1\right)}{\Gamma\left(\nu-\mu+% 2\right)},}}
    0 references
    DLMF id
    DLMF:14.2.E5
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2abdec
    0 references
    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.m2aadec
    0 references
    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1ddec
    0 references
    Q11591
    Defining formula
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C14.S1.XMD7.m1cdec
    0 references
    Q11590
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1ddec
    0 references
     
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