DLMF:18.17.E10 (Q5751)

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DLMF:18.17.E10
No description defined

    Statements

    Defining formula
    x β + μ ⁢ ( x + 1 ) n Γ ⁡ ( β + μ + n + 1 ) ⁢ P n ( α , β + μ ) ⁡ ( x - 1 x + 1 ) = ∫ 0 x y β ⁢ ( y + 1 ) n Γ ⁡ ( β + n + 1 ) ⁢ P n ( α , β ) ⁡ ( y - 1 y + 1 ) ⁢ ( x - y ) μ - 1 Γ ⁡ ( μ ) ⁢ d y , superscript 𝑥 𝛽 𝜇 superscript 𝑥 1 𝑛 Euler-Gamma 𝛽 𝜇 𝑛 1 Jacobi-polynomial-P 𝛼 𝛽 𝜇 𝑛 𝑥 1 𝑥 1 superscript subscript 0 𝑥 superscript 𝑦 𝛽 superscript 𝑦 1 𝑛 Euler-Gamma 𝛽 𝑛 1 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑦 1 𝑦 1 superscript 𝑥 𝑦 𝜇 1 Euler-Gamma 𝜇 𝑦 {\displaystyle{\displaystyle\frac{x^{\beta+\mu}(x+1)^{n}}{\Gamma\left(\beta+% \mu+n+1\right)}P^{(\alpha,\beta+\mu)}_{n}\left(\frac{x-1}{x+1}\right)=\int_{0}% ^{x}\frac{y^{\beta}(y+1)^{n}}{\Gamma\left(\beta+n+1\right)}P^{(\alpha,\beta)}_% {n}\left(\frac{y-1}{y+1}\right)\*\frac{(x-y)^{\mu-1}}{\Gamma\left(\mu\right)}% \mathrm{d}y,}}
    0 references
    DLMF id
    DLMF:18.17.E10
    0 references
    constraint
    x > 0 𝑥 0 {\displaystyle{\displaystyle x>0}}
    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.m2abdec
    0 references
    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.m2abdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aidec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aidec
    0 references
    Q11743
    Defining formula
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C18.S1.XMD1.m1edec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1idec
    0 references
    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1gdec
    0 references
     
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