DLMF:18.10.E4 (Q5628): Difference between revisions

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Property / constraint
 

α > - 1 2 𝛼 1 2 {\displaystyle{\displaystyle\alpha>-\frac{1}{2}}}

\alpha>-\frac{1}{2}
Property / constraint: α > - 1 2 𝛼 1 2 {\displaystyle{\displaystyle\alpha>-\frac{1}{2}}} / rank
 
Normal rank

Revision as of 16:59, 30 December 2019

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DLMF:18.10.E4
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    Statements

    test
    P n ( α , α ) ⁡ ( cos ⁡ θ ) P n ( α , α ) ⁡ ( 1 ) = C n ( α + 1 2 ) ⁡ ( cos ⁡ θ ) C n ( α + 1 2 ) ⁡ ( 1 ) = Γ ⁡ ( α + 1 ) π 1 2 ⁢ Γ ⁡ ( α + 1 2 ) ⁢ ∫ 0 π ( cos ⁡ θ + i ⁢ sin ⁡ θ ⁢ cos ⁡ ϕ ) n ⁢ ( sin ⁡ ϕ ) 2 ⁢ α ⁢ d ϕ , Jacobi-polynomial-P 𝛼 𝛼 𝑛 𝜃 Jacobi-polynomial-P 𝛼 𝛼 𝑛 1 ultraspherical-Gegenbauer-polynomial 𝛼 1 2 𝑛 𝜃 ultraspherical-Gegenbauer-polynomial 𝛼 1 2 𝑛 1 Euler-Gamma 𝛼 1 superscript 𝜋 1 2 Euler-Gamma 𝛼 1 2 superscript subscript 0 𝜋 superscript 𝜃 𝑖 𝜃 italic-ϕ 𝑛 superscript italic-ϕ 2 𝛼 italic-ϕ {\displaystyle{\displaystyle{\frac{P^{(\alpha,\alpha)}_{n}\left(\cos\theta% \right)}{P^{(\alpha,\alpha)}_{n}\left(1\right)}=\frac{C^{(\alpha+\frac{1}{2})}% _{n}\left(\cos\theta\right)}{C^{(\alpha+\frac{1}{2})}_{n}\left(1\right)}}=% \frac{\Gamma\left(\alpha+1\right)}{\pi^{\frac{1}{2}}\Gamma{(\alpha+\tfrac{1}{2% })}}\*{\int_{0}^{\pi}(\cos\theta+i\sin\theta\cos\phi)^{n}\*(\sin\phi)^{2\alpha% }\mathrm{d}\phi},}}
    0 references
    DLMF id
    DLMF:18.10.E4
    0 references
    constraint
    α > - 1 2 𝛼 1 2 {\displaystyle{\displaystyle\alpha>-\frac{1}{2}}}
    0 references
     
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