DLMF:14.3.E20 (Q4709)

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DLMF:14.3.E20
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    test
    2 ⁢ sin ⁡ ( μ ⁢ π ) π ⁢ 𝑸 ν μ ⁡ ( x ) = ( x + 1 ) μ / 2 Γ ⁡ ( ν + μ + 1 ) ⁢ ( x - 1 ) μ / 2 ⁢ 𝐅 ⁡ ( ν + 1 , - ν ; 1 - μ ; 1 2 - 1 2 ⁢ x ) - ( x - 1 ) μ / 2 Γ ⁡ ( ν - μ + 1 ) ⁢ ( x + 1 ) μ / 2 ⁢ 𝐅 ⁡ ( ν + 1 , - ν ; μ + 1 ; 1 2 - 1 2 ⁢ x ) . 2 𝜇 𝜋 𝜋 associated-Legendre-black-Q 𝜇 𝜈 𝑥 superscript 𝑥 1 𝜇 2 Euler-Gamma 𝜈 𝜇 1 superscript 𝑥 1 𝜇 2 scaled-hypergeometric-bold-F 𝜈 1 𝜈 1 𝜇 1 2 1 2 𝑥 superscript 𝑥 1 𝜇 2 Euler-Gamma 𝜈 𝜇 1 superscript 𝑥 1 𝜇 2 scaled-hypergeometric-bold-F 𝜈 1 𝜈 𝜇 1 1 2 1 2 𝑥 {\displaystyle{\displaystyle\frac{2\sin\left(\mu\pi\right)}{\pi}\boldsymbol{Q}% ^{\mu}_{\nu}\left(x\right)=\frac{(x+1)^{\mu/2}}{\Gamma\left(\nu+\mu+1\right)(x% -1)^{\mu/2}}\mathbf{F}\left(\nu+1,-\nu;1-\mu;\tfrac{1}{2}-\tfrac{1}{2}x\right)% -\frac{(x-1)^{\mu/2}}{\Gamma\left(\nu-\mu+1\right)(x+1)^{\mu/2}}\mathbf{F}% \left(\nu+1,-\nu;\mu+1;\tfrac{1}{2}-\tfrac{1}{2}x\right).}}
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    DLMF id
    DLMF:14.3.E20
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    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aldec
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    Q11592
    test
    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.m3abdec
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    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2agdec
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    $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function
    test
    𝐅 ⁡ ( a , b ; c ; z ) scaled-hypergeometric-bold-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z% }\right)}}
    xml-id
    C15.S2.E2.m2apdec
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    sine function
    test
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2acdec
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    Q11589
    test
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1sdec
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    Q11591
    test
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C14.S1.XMD7.m1odec
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