DLMF:18.17.E13 (Q5754): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11727 / rank
 
Normal rank
Property / Symbols used: Q11727 / qualifier
 
Defining formula:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11727 / qualifier
 
xml-id: C18.S2.XMD3.m1jdec

Latest revision as of 15:47, 2 January 2020

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DLMF:18.17.E13
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    Statements

    Defining formula
    x 1 2 ⁢ n ⁢ ( x - 1 ) λ + μ - 1 2 Γ ⁡ ( λ + μ + 1 2 ) ⁢ C n ( λ + μ ) ⁡ ( x - 1 2 ) C n ( λ + μ ) ⁡ ( 1 ) = ∫ 1 x y 1 2 ⁢ n ⁢ ( y - 1 ) λ - 1 2 Γ ⁡ ( λ + 1 2 ) ⁢ C n ( λ ) ⁡ ( y - 1 2 ) C n ( λ ) ⁡ ( 1 ) ⁢ ( x - y ) μ - 1 Γ ⁡ ( μ ) ⁢ d y , superscript 𝑥 1 2 𝑛 superscript 𝑥 1 𝜆 𝜇 1 2 Euler-Gamma 𝜆 𝜇 1 2 ultraspherical-Gegenbauer-polynomial 𝜆 𝜇 𝑛 superscript 𝑥 1 2 ultraspherical-Gegenbauer-polynomial 𝜆 𝜇 𝑛 1 superscript subscript 1 𝑥 superscript 𝑦 1 2 𝑛 superscript 𝑦 1 𝜆 1 2 Euler-Gamma 𝜆 1 2 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 superscript 𝑦 1 2 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 1 superscript 𝑥 𝑦 𝜇 1 Euler-Gamma 𝜇 𝑦 {\displaystyle{\displaystyle\frac{x^{\frac{1}{2}n}(x-1)^{\lambda+\mu-\frac{1}{% 2}}}{\Gamma\left(\lambda+\mu+\tfrac{1}{2}\right)}\frac{C^{(\lambda+\mu)}_{n}% \left(x^{-\frac{1}{2}}\right)}{C^{(\lambda+\mu)}_{n}\left(1\right)}=\int_{1}^{% x}\frac{y^{\frac{1}{2}n}(y-1)^{\lambda-\frac{1}{2}}}{\Gamma\left(\lambda+% \tfrac{1}{2}\right)}\frac{C^{(\lambda)}_{n}\left(y^{-\frac{1}{2}}\right)}{C^{(% \lambda)}_{n}\left(1\right)}\frac{(x-y)^{\mu-1}}{\Gamma\left(\mu\right)}% \mathrm{d}y,}}
    0 references
    DLMF id
    DLMF:18.17.E13
    0 references
    constraint
    x > 1 𝑥 1 {\displaystyle{\displaystyle x>1}}
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aedec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aldec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aldec
    0 references
    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.m2abdec
    0 references
    Q11743
    Defining formula
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C18.S1.XMD1.m1hdec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1ldec
    0 references
    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1jdec
    0 references
     
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