DLMF:18.17.E37 (Q5778): Difference between revisions

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Property / constraint
 

ℜ ⁡ z > 0 𝑧 0 {\displaystyle{\displaystyle\Re z>0}}

\realpart@@{z}>0
Property / constraint: ℜ ⁡ z > 0 𝑧 0 {\displaystyle{\displaystyle\Re z>0}} / rank
 
Normal rank

Revision as of 17:02, 30 December 2019

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DLMF:18.17.E37
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    Statements

    test
    ∫ 0 1 ( 1 - x 2 ) λ - 1 2 ⁢ C n ( λ ) ⁡ ( x ) ⁢ x z - 1 ⁢ d x = π ⁢  2 1 - 2 ⁢ λ - z ⁢ Γ ⁡ ( n + 2 ⁢ λ ) ⁢ Γ ⁡ ( z ) n ! ⁢ Γ ⁡ ( λ ) ⁢ Γ ⁡ ( 1 2 + 1 2 ⁢ n + λ + 1 2 ⁢ z ) ⁢ Γ ⁡ ( 1 2 + 1 2 ⁢ z - 1 2 ⁢ n ) , superscript subscript 0 1 superscript 1 superscript 𝑥 2 𝜆 1 2 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 superscript 𝑥 𝑧 1 𝑥 𝜋 superscript  2 1 2 𝜆 𝑧 Euler-Gamma 𝑛 2 𝜆 Euler-Gamma 𝑧 𝑛 Euler-Gamma 𝜆 Euler-Gamma 1 2 1 2 𝑛 𝜆 1 2 𝑧 Euler-Gamma 1 2 1 2 𝑧 1 2 𝑛 {\displaystyle{\displaystyle\int_{0}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}C^{(% \lambda)}_{n}\left(x\right)x^{z-1}\mathrm{d}x=\frac{\pi\,2^{1-2\lambda-z}% \Gamma\left(n+2\lambda\right)\Gamma\left(z\right)}{n!\Gamma\left(\lambda\right% )\Gamma\left(\frac{1}{2}+\frac{1}{2}n+\lambda+\frac{1}{2}z\right)\Gamma\left(% \frac{1}{2}+\frac{1}{2}z-\frac{1}{2}n\right)},}}
    0 references
    DLMF id
    DLMF:18.17.E37
    0 references
    constraint
    ℜ ⁡ z > 0 𝑧 0 {\displaystyle{\displaystyle\Re z>0}}
    0 references
     
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