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DLMF:18.7.E26 - MaRDI portal

DLMF:18.7.E26 (Q6291)

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DLMF:18.7.E26
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    Statements

    DLMF defining formula
    lim α → ∞ ⁡ ( 2 α ) 1 2 ⁢ n ⁢ L n ( α ) ⁡ ( ( 2 ⁢ α ) 1 2 ⁢ x + α ) = ( - 1 ) n n ! ⁢ H n ⁡ ( x ) . subscript → 𝛼 superscript 2 𝛼 1 2 𝑛 Laguerre-polynomial-L 𝛼 𝑛 superscript 2 𝛼 1 2 𝑥 𝛼 superscript 1 𝑛 𝑛 Hermite-polynomial-H 𝑛 𝑥 {\displaystyle{\displaystyle\lim_{\alpha\to\infty}\left(\frac{2}{\alpha}\right% )^{\frac{1}{2}n}L^{(\alpha)}_{n}\left((2\alpha)^{\frac{1}{2}}x+\alpha\right)=% \frac{(-1)^{n}}{n!}H_{n}\left(x\right).}}
    0 references
    Symbols used
    Hermite polynomial
    DLMF defining formula
    H n ⁡ ( x ) Hermite-polynomial-H 𝑛 𝑥 {\displaystyle{\displaystyle H_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C18.S3.T1.t1.r13.m2afdec
    0 references
    Laguerre (or generalized Laguerre) polynomial
    DLMF 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.m2addec
    0 references
    Q10729
    DLMF defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5addec
    0 references
    nonnegative integer
    DLMF defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1ydec
    0 references
    real variable
    DLMF defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1ydec
    0 references

    Identifiers

     
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