DLMF:18.17.E9 (Q5750)

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DLMF:18.17.E9
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    Statements

    Defining formula
    ( 1 - x ) α + μ ⁢ P n ( α + μ , β - μ ) ⁡ ( x ) Γ ⁡ ( α + μ + n + 1 ) = ∫ x 1 ( 1 - y ) α ⁢ P n ( α , β ) ⁡ ( y ) Γ ⁡ ( α + n + 1 ) ⁢ ( y - x ) μ - 1 Γ ⁡ ( μ ) ⁢ d y , superscript 1 𝑥 𝛼 𝜇 Jacobi-polynomial-P 𝛼 𝜇 𝛽 𝜇 𝑛 𝑥 Euler-Gamma 𝛼 𝜇 𝑛 1 superscript subscript 𝑥 1 superscript 1 𝑦 𝛼 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑦 Euler-Gamma 𝛼 𝑛 1 superscript 𝑦 𝑥 𝜇 1 Euler-Gamma 𝜇 𝑦 {\displaystyle{\displaystyle\frac{(1-x)^{\alpha+\mu}P^{(\alpha+\mu,\beta-\mu)}% _{n}\left(x\right)}{\Gamma\left(\alpha+\mu+n+1\right)}=\int_{x}^{1}\frac{(1-y)% ^{\alpha}P^{(\alpha,\beta)}_{n}\left(y\right)}{\Gamma\left(\alpha+n+1\right)}% \frac{(y-x)^{\mu-1}}{\Gamma\left(\mu\right)}\mathrm{d}y,}}
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    DLMF id
    DLMF:18.17.E9
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    constraint
    - 1 < x < 1 1 𝑥 1 {\displaystyle{\displaystyle-1<x<1}}
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