DLMF:14.9.E12 (Q4804)

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DLMF:14.9.E12
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    Statements

    Defining formula
    cos ⁡ ( ν ⁢ π ) ⁢ P ν - μ ⁡ ( x ) = - 𝑸 ν μ ⁡ ( x ) Γ ⁡ ( μ - ν ) + 𝑸 - ν - 1 μ ⁡ ( x ) Γ ⁡ ( ν + μ + 1 ) . 𝜈 𝜋 Legendre-P-first-kind 𝜇 𝜈 𝑥 associated-Legendre-black-Q 𝜇 𝜈 𝑥 Euler-Gamma 𝜇 𝜈 associated-Legendre-black-Q 𝜇 𝜈 1 𝑥 Euler-Gamma 𝜈 𝜇 1 {\displaystyle{\displaystyle\cos\left(\nu\pi\right)P^{-\mu}_{\nu}\left(x\right% )=-\frac{\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)}{\Gamma\left(\mu-\nu\right)}% +\frac{\boldsymbol{Q}^{\mu}_{-\nu-1}\left(x\right)}{\Gamma\left(\nu+\mu+1% \right)}.}}
    0 references
    DLMF id
    DLMF:14.9.E12
    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.m2afdec
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    Q11481
    Defining formula
    P ν μ ⁡ ( z ) Legendre-P-first-kind 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C14.S21.SS1.p1.m1aadec
    0 references
    Q11592
    Defining formula
    𝑸 ν μ ⁡ ( z ) associated-Legendre-black-Q 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(% \NVar{z}\right)}}
    xml-id
    C14.S21.SS1.p1.m3adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2agdec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2afdec
    0 references
    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1kdec
    0 references
    Q11591
    Defining formula
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C14.S1.XMD7.m1idec
    0 references
     
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