DLMF:18.17.E39 (Q5780)

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DLMF:18.17.E39
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    Statements

    Defining formula
    ∫ 0 1 P 2 ⁢ n + 1 ⁡ ( x ) ⁢ x z - 1 ⁢ d x = ( - 1 ) n ⁢ ( 1 - 1 2 ⁢ z ) n 2 ⁢ ( 1 2 + 1 2 ⁢ z ) n + 1 , superscript subscript 0 1 Legendre-spherical-polynomial 2 𝑛 1 𝑥 superscript 𝑥 𝑧 1 𝑥 superscript 1 𝑛 Pochhammer 1 1 2 𝑧 𝑛 2 Pochhammer 1 2 1 2 𝑧 𝑛 1 {\displaystyle{\displaystyle\int_{0}^{1}P_{2n+1}\left(x\right)x^{z-1}\mathrm{d% }x=\frac{(-1)^{n}{\left(1-\frac{1}{2}z\right)_{n}}}{2{\left(\frac{1}{2}+\frac{% 1}{2}z\right)_{n+1}}},}}
    0 references
    DLMF id
    DLMF:18.17.E39
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    constraint
    ℜ ⁡ z > - 1 𝑧 1 {\displaystyle{\displaystyle\Re z>-1}}
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    Symbols used
    Q10920
    Defining formula
    P n ⁡ ( x ) Legendre-spherical-polynomial 𝑛 𝑥 {\displaystyle{\displaystyle P_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C18.S3.T1.t1.r10.m2afdec
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    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1abdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aaldec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aaldec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1addec
    0 references
    Q11750
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C18.S1.XMD2.m1fdec
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    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1aldec
    0 references
    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1ajdec
    0 references
     
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