DLMF:18.17.E36 (Q5777)

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DLMF:18.17.E36
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    Statements

    Defining formula
    ∫ - 1 1 ( 1 - x ) z - 1 ⁢ ( 1 + x ) β ⁢ P n ( α , β ) ⁡ ( x ) ⁢ d x = 2 β + z ⁢ Γ ⁡ ( z ) ⁢ Γ ⁡ ( 1 + β + n ) ⁢ ( 1 + α - z ) n n ! ⁢ Γ ⁡ ( 1 + β + z + n ) , superscript subscript 1 1 superscript 1 𝑥 𝑧 1 superscript 1 𝑥 𝛽 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 𝑥 superscript 2 𝛽 𝑧 Euler-Gamma 𝑧 Euler-Gamma 1 𝛽 𝑛 Pochhammer 1 𝛼 𝑧 𝑛 𝑛 Euler-Gamma 1 𝛽 𝑧 𝑛 {\displaystyle{\displaystyle\int_{-1}^{1}(1-x)^{z-1}(1+x)^{\beta}P^{(\alpha,% \beta)}_{n}\left(x\right)\mathrm{d}x=\frac{2^{\beta+z}\Gamma\left(z\right)% \Gamma\left(1+\beta+n\right){\left(1+\alpha-z\right)_{n}}}{n!\Gamma\left(1+% \beta+z+n\right)},}}
    0 references
    DLMF id
    DLMF:18.17.E36
    0 references
    constraint
    ℜ ⁡ z > 0 𝑧 0 {\displaystyle{\displaystyle\Re z>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.m2andec
    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.m2afdec
    0 references
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aaidec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5akdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aaidec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aadec
    0 references
    Q11750
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C18.S1.XMD2.m1cdec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1aidec
    0 references
    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1agdec
    0 references
     
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