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DLMF:18.7.E20 - MaRDI portal

DLMF:18.7.E20 (Q6285)

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DLMF:18.7.E20
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    Statements

    DLMF defining formula
    H 2 ⁢ n + 1 ⁡ ( x ) = ( - 1 ) n ⁢ 2 2 ⁢ n + 1 ⁢ n ! ⁢ x ⁢ L n ( 1 2 ) ⁡ ( x 2 ) . Hermite-polynomial-H 2 𝑛 1 𝑥 superscript 1 𝑛 superscript 2 2 𝑛 1 𝑛 𝑥 Laguerre-polynomial-L 1 2 𝑛 superscript 𝑥 2 {\displaystyle{\displaystyle H_{2n+1}\left(x\right)=(-1)^{n}2^{2n+1}n!\,xL^{(% \frac{1}{2})}_{n}\left(x^{2}\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.m2acdec
    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.m2aadec
    0 references
    Q10729
    DLMF defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5aadec
    0 references
    nonnegative integer
    DLMF defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1sdec
    0 references
    real variable
    DLMF defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1sdec
    0 references

    Identifiers

     
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