DLMF:18.11.E4 (Q5638)

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DLMF:18.11.E4
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    Statements

    Defining formula
    𝐻𝑒 n ⁡ ( x ) = 2 1 2 ⁢ n ⁢ U ⁡ ( - 1 2 ⁢ n , 1 2 , 1 2 ⁢ x 2 ) = 2 1 2 ⁢ ( n - 1 ) ⁢ x ⁢ U ⁡ ( - 1 2 ⁢ n + 1 2 , 3 2 , 1 2 ⁢ x 2 ) = e 1 4 ⁢ x 2 ⁢ U ⁡ ( - n - 1 2 , x ) . Hermite-polynomial-He 𝑛 𝑥 superscript 2 1 2 𝑛 Kummer-confluent-hypergeometric-U 1 2 𝑛 1 2 1 2 superscript 𝑥 2 superscript 2 1 2 𝑛 1 𝑥 Kummer-confluent-hypergeometric-U 1 2 𝑛 1 2 3 2 1 2 superscript 𝑥 2 superscript 𝑒 1 4 superscript 𝑥 2 parabolic-U 𝑛 1 2 𝑥 {\displaystyle{\displaystyle\mathit{He}_{n}\left(x\right)=2^{\frac{1}{2}n}U% \left(-\tfrac{1}{2}n,\tfrac{1}{2},\tfrac{1}{2}x^{2}\right)=2^{\frac{1}{2}(n-1)% }xU\left(-\tfrac{1}{2}n+\tfrac{1}{2},\tfrac{3}{2},\tfrac{1}{2}x^{2}\right)=e^{% \tfrac{1}{4}x^{2}}U\left(-n-\tfrac{1}{2},x\right).}}
    0 references
    DLMF id
    DLMF:18.11.E4
    0 references
    Symbols used
    Q11525
    Defining formula
    𝐻𝑒 n ⁡ ( x ) Hermite-polynomial-He 𝑛 𝑥 {\displaystyle{\displaystyle\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C18.S3.T1.t1.r14.m2adec
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    Kummer confluent hypergeometric function
    Defining formula
    U ⁡ ( a , b , z ) Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{b},\NVar{z}\right)}}
    xml-id
    C13.S2.E6.m2abdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
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    Q11320
    Defining formula
    U ⁡ ( a , z ) parabolic-U 𝑎 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C12.S2.SS1.p1.m1aadec
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    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1cdec
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    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1cdec
    0 references
     
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