DLMF:18.17.E12 (Q5753)

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DLMF:18.17.E12
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    Statements

    Defining formula
    Γ ⁡ ( λ - μ ) ⁢ C n ( λ - μ ) ⁡ ( x - 1 2 ) x λ - μ + 1 2 ⁢ n = ∫ x ∞ Γ ⁡ ( λ ) ⁢ C n ( λ ) ⁡ ( y - 1 2 ) y λ + 1 2 ⁢ n ⁢ ( y - x ) μ - 1 Γ ⁡ ( μ ) ⁢ d y , Euler-Gamma 𝜆 𝜇 ultraspherical-Gegenbauer-polynomial 𝜆 𝜇 𝑛 superscript 𝑥 1 2 superscript 𝑥 𝜆 𝜇 1 2 𝑛 superscript subscript 𝑥 Euler-Gamma 𝜆 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 superscript 𝑦 1 2 superscript 𝑦 𝜆 1 2 𝑛 superscript 𝑦 𝑥 𝜇 1 Euler-Gamma 𝜇 𝑦 {\displaystyle{\displaystyle\frac{\Gamma\left(\lambda-\mu\right)C^{(\lambda-% \mu)}_{n}\left(x^{-\frac{1}{2}}\right)}{x^{\lambda-\mu+\frac{1}{2}n}}=\int_{x}% ^{\infty}\frac{\Gamma\left(\lambda\right)C^{(\lambda)}_{n}\left(y^{-\frac{1}{2% }}\right)}{y^{\lambda+\frac{1}{2}n}}\frac{(y-x)^{\mu-1}}{\Gamma\left(\mu\right% )}\mathrm{d}y,}}
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    DLMF id
    DLMF:18.17.E12
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    constraint
    x > 0 𝑥 0 {\displaystyle{\displaystyle x>0}}
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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.m2addec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1akdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3akdec
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    Q11475
    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.m2aadec
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