DLMF:18.18.E27 (Q5817)

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DLMF:18.18.E27
No description defined

    Statements

    Defining formula
    ∑ n = 0 ∞ n ! ⁢ L n ( α ) ⁡ ( x ) ⁢ L n ( α ) ⁡ ( y ) ( α + 1 ) n ⁢ z n = Γ ⁡ ( α + 1 ) ⁢ ( x ⁢ y ⁢ z ) - 1 2 ⁢ α 1 - z ⁢ exp ⁡ ( - ( x + y ) ⁢ z 1 - z ) ⁢ I α ⁡ ( 2 ⁢ ( x ⁢ y ⁢ z ) 1 2 1 - z ) , superscript subscript 𝑛 0 𝑛 Laguerre-polynomial-L 𝛼 𝑛 𝑥 Laguerre-polynomial-L 𝛼 𝑛 𝑦 Pochhammer 𝛼 1 𝑛 superscript 𝑧 𝑛 Euler-Gamma 𝛼 1 superscript 𝑥 𝑦 𝑧 1 2 𝛼 1 𝑧 𝑥 𝑦 𝑧 1 𝑧 modified-Bessel-first-kind 𝛼 2 superscript 𝑥 𝑦 𝑧 1 2 1 𝑧 {\displaystyle{\displaystyle\sum_{n=0}^{\infty}\frac{n!\,L^{(\alpha)}_{n}\left% (x\right)L^{(\alpha)}_{n}\left(y\right)}{{\left(\alpha+1\right)_{n}}}z^{n}=% \frac{\Gamma\left(\alpha+1\right)(xyz)^{-\frac{1}{2}\alpha}}{1-z}\*\exp\left(% \frac{-(x+y)z}{1-z}\right)I_{\alpha}\left(\frac{2(xyz)^{\frac{1}{2}}}{1-z}% \right),}}
    0 references
    DLMF id
    DLMF:18.18.E27
    0 references
    constraint
    | z | < 1 𝑧 1 {\displaystyle{\displaystyle|z|<1}}
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    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2abdec
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    Q11333
    Defining formula
    L n ( α ) ⁡ ( x ) Laguerre-polynomial-L 𝛼 𝑛 𝑥 {\displaystyle{\displaystyle L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}% \right)}}
    xml-id
    C18.S3.T1.t1.r12.m2afdec
    0 references
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1akdec
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    exponential function
    Defining formula
    exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}
    xml-id
    C4.S2.E19.m2adec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5andec
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    modified Bessel function of the first kind
    Defining formula
    I ν ⁡ ( z ) modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E2.m2adec
    0 references
    Q11743
    Defining formula
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C18.S1.XMD1.m1adec
    0 references
    Q11750
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C18.S1.XMD2.m1adec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1zdec
    0 references
    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1vdec
    0 references
     
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