DLMF:18.12.E14 (Q5656)

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DLMF:18.12.E14
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    Statements

    test
    Γ ⁡ ( α + 1 ) ⁢ ( x ⁢ z ) - 1 2 ⁢ α ⁢ e z ⁢ J α ⁡ ( 2 ⁢ x ⁢ z ) = ∑ n = 0 ∞ L n ( α ) ⁡ ( x ) ( α + 1 ) n ⁢ z n . Euler-Gamma 𝛼 1 superscript 𝑥 𝑧 1 2 𝛼 superscript 𝑒 𝑧 Bessel-J 𝛼 2 𝑥 𝑧 superscript subscript 𝑛 0 Laguerre-polynomial-L 𝛼 𝑛 𝑥 Pochhammer 𝛼 1 𝑛 superscript 𝑧 𝑛 {\displaystyle{\displaystyle\Gamma\left(\alpha+1\right)(xz)^{-\frac{1}{2}% \alpha}e^{z}J_{\alpha}\left(2\sqrt{xz}\right)=\sum_{n=0}^{\infty}\frac{L^{(% \alpha)}_{n}\left(x\right)}{{\left(\alpha+1\right)_{n}}}z^{n}.}}
    0 references
    DLMF id
    DLMF:18.12.E14
    0 references
    Symbols used
    Bessel function of the first kind
    test
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2acdec
    0 references
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2abdec
    0 references
    Q11333
    test
    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.m2aadec
    0 references
    Q10759
    test
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1acdec
    0 references
    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
    0 references
    Q11750
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C18.S1.XMD2.m1mdec
    0 references
     
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