DLMF:18.10.E3 (Q5627)

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DLMF:18.10.E3
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    Statements

    Defining formula
    P n ( α , β ) ⁡ ( cos ⁡ θ ) P n ( α , β ) ⁡ ( 1 ) = 2 ⁢ Γ ⁡ ( α + 1 ) π 1 2 ⁢ Γ ⁡ ( α - β ) ⁢ Γ ⁡ ( β + 1 2 ) ⁢ ∫ 0 1 ∫ 0 π ( ( cos ⁡ 1 2 ⁢ θ ) 2 - r 2 ⁢ ( sin ⁡ 1 2 ⁢ θ ) 2 + i ⁢ r ⁢ sin ⁡ θ ⁢ cos ⁡ ϕ ) n ⁢ ( 1 - r 2 ) α - β - 1 ⁢ r 2 ⁢ β + 1 ⁢ ( sin ⁡ ϕ ) 2 ⁢ β ⁢ d ϕ ⁢ d r , Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝜃 Jacobi-polynomial-P 𝛼 𝛽 𝑛 1 2 Euler-Gamma 𝛼 1 superscript 𝜋 1 2 Euler-Gamma 𝛼 𝛽 Euler-Gamma 𝛽 1 2 superscript subscript 0 1 superscript subscript 0 𝜋 superscript superscript 1 2 𝜃 2 superscript 𝑟 2 superscript 1 2 𝜃 2 𝑖 𝑟 𝜃 italic-ϕ 𝑛 superscript 1 superscript 𝑟 2 𝛼 𝛽 1 superscript 𝑟 2 𝛽 1 superscript italic-ϕ 2 𝛽 italic-ϕ 𝑟 {\displaystyle{\displaystyle\frac{P^{(\alpha,\beta)}_{n}\left(\cos\theta\right% )}{P^{(\alpha,\beta)}_{n}\left(1\right)}=\frac{2\Gamma\left(\alpha+1\right)}{% \pi^{\frac{1}{2}}\Gamma\left(\alpha-\beta\right)\Gamma\left(\beta+\tfrac{1}{2}% \right)}\*\int_{0}^{1}\int_{0}^{\pi}\left((\cos\tfrac{1}{2}\theta)^{2}-r^{2}(% \sin\tfrac{1}{2}\theta)^{2}+ir\sin\theta\cos\phi\right)^{n}(1-r^{2})^{\alpha-% \beta-1}r^{2\beta+1}(\sin\phi)^{2\beta}\mathrm{d}\phi\mathrm{d}r,}}
    0 references
    DLMF id
    DLMF:18.10.E3
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    constraint
    α > β > - 1 2 𝛼 𝛽 1 2 {\displaystyle{\displaystyle\alpha>\beta>-\tfrac{1}{2}}}
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    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aadec
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    Q11660
    Defining formula
    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)}}
    xml-id
    C18.S3.T1.t1.r2.m2aadec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2abdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abdec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abdec
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