DLMF:18.17.E11 (Q5752)

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DLMF:18.17.E11
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    Γ ( n + α + β - μ + 1 ) x n + α + β - μ + 1 P n ( α , β - μ ) ( 1 - 2 x - 1 ) = x Γ ( n + α + β + 1 ) y n + α + β + 1 P n ( α , β ) ( 1 - 2 y - 1 ) ( y - x ) μ - 1 Γ ( μ ) d y , Euler-Gamma 𝑛 𝛼 𝛽 𝜇 1 superscript 𝑥 𝑛 𝛼 𝛽 𝜇 1 Jacobi-polynomial-P 𝛼 𝛽 𝜇 𝑛 1 2 superscript 𝑥 1 superscript subscript 𝑥 Euler-Gamma 𝑛 𝛼 𝛽 1 superscript 𝑦 𝑛 𝛼 𝛽 1 Jacobi-polynomial-P 𝛼 𝛽 𝑛 1 2 superscript 𝑦 1 superscript 𝑦 𝑥 𝜇 1 Euler-Gamma 𝜇 𝑦 {\displaystyle{\displaystyle\frac{\Gamma\left(n+\alpha+\beta-\mu+1\right)}{x^{% n+\alpha+\beta-\mu+1}}P^{(\alpha,\beta-\mu)}_{n}\left(1-2x^{-1}\right)=\int_{x% }^{\infty}\frac{\Gamma\left(n+\alpha+\beta+1\right)}{y^{n+\alpha+\beta+1}}P^{(% \alpha,\beta)}_{n}\left(1-2y^{-1}\right)\*\frac{(y-x)^{\mu-1}}{\Gamma\left(\mu% \right)}\mathrm{d}y,}}
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    DLMF:18.17.E11
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    x > 1 𝑥 1 {\displaystyle{\displaystyle x>1}}
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    Γ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    C5.S2.E1.m2acdec
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    P n ( α , β ) ( x ) Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 {\displaystyle{\displaystyle P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(% \NVar{x}\right)}}
    C18.S3.T1.t1.r2.m2acdec
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    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    C1.S4.SS4.m1ajdec
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    {\displaystyle{\displaystyle\int}}
    C1.S4.SS4.m3ajdec
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    y 𝑦 {\displaystyle{\displaystyle y}}
    C18.S1.XMD1.m1fdec
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    n 𝑛 {\displaystyle{\displaystyle n}}
    C18.S1.XMD6.m1jdec
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