DLMF:18.17.E8 (Q5749): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q10922 / rank
 
Normal rank
Property / Symbols used: Q10922 / qualifier
 
Defining formula:

H n ⁡ ( x ) Hermite-polynomial-H 𝑛 𝑥 {\displaystyle{\displaystyle H_{\NVar{n}}\left(\NVar{x}\right)}}

\HermitepolyH{\NVar{n}}@{\NVar{x}}
Property / Symbols used: Q10922 / qualifier
 
xml-id: C18.S3.T1.t1.r13.m2abdec

Revision as of 15:45, 2 January 2020

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DLMF:18.17.E8
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    Statements

    Defining formula
    ( H n ⁡ ( x ) ) 2 + 2 n ⁢ ( n ! ) 2 ⁢ e x 2 ⁢ ( V ⁡ ( - n - 1 2 , 2 1 2 ⁢ x ) ) 2 = 2 n + 3 2 ⁢ n ! ⁢ e x 2 π ⁢ ∫ 0 ∞ e - ( 2 ⁢ n + 1 ) ⁢ t + x 2 ⁢ tanh ⁡ t ( sinh ⁡ 2 ⁢ t ) 1 2 ⁢ d t . superscript Hermite-polynomial-H 𝑛 𝑥 2 superscript 2 𝑛 superscript 𝑛 2 superscript 𝑒 superscript 𝑥 2 superscript parabolic-V 𝑛 1 2 superscript 2 1 2 𝑥 2 superscript 2 𝑛 3 2 𝑛 superscript 𝑒 superscript 𝑥 2 𝜋 superscript subscript 0 superscript 𝑒 2 𝑛 1 𝑡 superscript 𝑥 2 𝑡 superscript 2 𝑡 1 2 𝑡 {\displaystyle{\displaystyle\left(H_{n}\left(x\right)\right)^{2}+2^{n}(n!)^{2}% e^{x^{2}}\left(V\left(-n-\tfrac{1}{2},2^{\frac{1}{2}}x\right)\right)^{2}=\frac% {2^{n+\frac{3}{2}}n!\,e^{x^{2}}}{\pi}\int_{0}^{\infty}\frac{e^{-(2n+1)t+x^{2}% \tanh t}}{(\sinh 2t)^{\frac{1}{2}}}\mathrm{d}t.}}
    0 references
    DLMF id
    DLMF:18.17.E8
    0 references
    Symbols used
    Q10922
    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.m2abdec
    0 references
     
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