DLMF:14.2.E11 (Q4689)

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DLMF:14.2.E11
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    Statements

    test
    P ν + 1 μ ⁡ ( x ) ⁢ Q ν μ ⁡ ( x ) - P ν μ ⁡ ( x ) ⁢ Q ν + 1 μ ⁡ ( x ) = e μ ⁢ π ⁢ i ⁢ Γ ⁡ ( ν + μ + 1 ) Γ ⁡ ( ν - μ + 2 ) . Legendre-P-first-kind 𝜇 𝜈 1 𝑥 Legendre-Q-second-kind 𝜇 𝜈 𝑥 Legendre-P-first-kind 𝜇 𝜈 𝑥 Legendre-Q-second-kind 𝜇 𝜈 1 𝑥 superscript 𝑒 𝜇 𝜋 𝑖 Euler-Gamma 𝜈 𝜇 1 Euler-Gamma 𝜈 𝜇 2 {\displaystyle{\displaystyle P^{\mu}_{\nu+1}\left(x\right)Q^{\mu}_{\nu}\left(x% \right)-P^{\mu}_{\nu}\left(x\right)Q^{\mu}_{\nu+1}\left(x\right)=e^{\mu\pi i}% \frac{\Gamma\left(\nu+\mu+1\right)}{\Gamma\left(\nu-\mu+2\right)}.}}
    0 references
    DLMF id
    DLMF:14.2.E11
    0 references
    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aedec
    0 references
    Q11481
    test
    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.m1acdec
    0 references
    Q11470
    test
    Q ν μ ⁡ ( z ) Legendre-Q-second-kind 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle Q^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C14.S21.SS1.p1.m2aadec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2addec
    0 references
    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
    0 references
    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aadec
    0 references
    Q11589
    test
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1jdec
    0 references
    Q11591
    test
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C14.S1.XMD7.m1idec
    0 references
    Q11590
    test
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1jdec
    0 references
     
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