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DLMF:18.17.E37 - MaRDI portal

DLMF:18.17.E37 (Q6475)

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DLMF:18.17.E37
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    Statements

    DLMF defining formula
    ∫ 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)},}}
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    constraint
    ℜ ⁡ z > 0 𝑧 0 {\displaystyle{\displaystyle\Re z>0}}
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    Symbols used
    gamma function
    DLMF defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aodec
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    the ratio of the circumference of a circle to its diameter
    DLMF defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2asdec
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    differential of x
    DLMF defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aajdec
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    Q10729
    DLMF defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5aldec
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    integral
    DLMF defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aajdec
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    real part
    DLMF defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1abdec
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    ultraspherical (or Gegenbauer) polynomial
    DLMF defining formula
    C n ( λ ) ⁡ ( x ) ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 {\displaystyle{\displaystyle C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}% \right)}}
    xml-id
    C18.S3.T1.t1.r3.m2aedec
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    complex variable
    DLMF defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C18.S1.XMD2.m1ddec
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    nonnegative integer
    DLMF defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1ajdec
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    real variable
    DLMF defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1ahdec
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    Identifiers

     
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