DLMF:18.17.E16 (Q5757)

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DLMF:18.17.E16
No description defined

    Statements

    Defining formula
    ∫ - 1 1 ( 1 - x ) α ⁢ ( 1 + x ) β ⁢ P n ( α , β ) ⁡ ( x ) ⁢ e i ⁢ x ⁢ y ⁢ d x = ( i ⁢ y ) n ⁢ e i ⁢ y n ! ⁢ 2 n + α + β + 1 ⁢ B ⁡ ( n + α + 1 , n + β + 1 ) ⁢ F 1 1 ⁡ ( n + α + 1 ; 2 ⁢ n + α + β + 2 ; - 2 ⁢ i ⁢ y ) . superscript subscript 1 1 superscript 1 𝑥 𝛼 superscript 1 𝑥 𝛽 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 superscript 𝑒 𝑖 𝑥 𝑦 𝑥 superscript 𝑖 𝑦 𝑛 superscript 𝑒 𝑖 𝑦 𝑛 superscript 2 𝑛 𝛼 𝛽 1 Euler-Beta 𝑛 𝛼 1 𝑛 𝛽 1 Kummer-confluent-hypergeometric-M-as-1F1 𝑛 𝛼 1 2 𝑛 𝛼 𝛽 2 2 𝑖 𝑦 {\displaystyle{\displaystyle\int_{-1}^{1}(1-x)^{\alpha}(1+x)^{\beta}P^{(\alpha% ,\beta)}_{n}\left(x\right)e^{ixy}\mathrm{d}x=\frac{(iy)^{n}e^{iy}}{n!}2^{n+% \alpha+\beta+1}\mathrm{B}\left(n+\alpha+1,n+\beta+1\right){{}_{1}F_{1}}\left(n% +\alpha+1;2n+\alpha+\beta+2;-2iy\right).}}
    0 references
    DLMF id
    DLMF:18.17.E16
    0 references
    Symbols used
    beta function
    Defining formula
    B ⁡ ( a , b ) Euler-Beta 𝑎 𝑏 {\displaystyle{\displaystyle\mathrm{B}\left(\NVar{a},\NVar{b}\right)}}
    xml-id
    C5.S12.E1.m2adec
    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.m2addec
    0 references
    Q11384
    Defining formula
    F 1 1 ⁡ ( a ; b ; z ) Kummer-confluent-hypergeometric-M-as-1F1 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle{{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}% \right)}}
    xml-id
    C16.S2.m7adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aodec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2acdec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5aadec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aodec
    0 references
    Q11743
    Defining formula
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C18.S1.XMD1.m1kdec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1odec
    0 references
    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1mdec
    0 references
     
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