DLMF:18.5.E13 (Q5521): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11726 / rank
 
Normal rank
Property / Symbols used: Q11726 / qualifier
 
Defining formula:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11726 / qualifier
 
xml-id: C18.S1.XMD6.m1mdec
Property / Symbols used
 
Property / Symbols used: Q11727 / rank
 
Normal rank
Property / Symbols used: Q11727 / qualifier
 
Defining formula:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11727 / qualifier
 
xml-id: C18.S2.XMD3.m1ldec

Latest revision as of 15:18, 2 January 2020

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DLMF:18.5.E13
No description defined

    Statements

    Defining formula
    H n ⁡ ( x ) = n ! ⁢ ∑ ℓ = 0 ⌊ n / 2 ⌋ ( - 1 ) ℓ ⁢ ( 2 ⁢ x ) n - 2 ⁢ ℓ ℓ ! ⁢ ( n - 2 ⁢ ℓ ) ! = ( 2 ⁢ x ) n ⁢ F 0 2 ⁡ ( - 1 2 ⁢ n , - 1 2 ⁢ n + 1 2 - ; - 1 x 2 ) . Hermite-polynomial-H 𝑛 𝑥 𝑛 superscript subscript ℓ 0 𝑛 2 superscript 1 ℓ superscript 2 𝑥 𝑛 2 ℓ ℓ 𝑛 2 ℓ superscript 2 𝑥 𝑛 Gauss-hypergeometric-pFq 2 0 1 2 𝑛 1 2 𝑛 1 2 1 superscript 𝑥 2 {\displaystyle{\displaystyle H_{n}\left(x\right)=n!\sum_{\ell=0}^{\left\lfloor n% /2\right\rfloor}\frac{(-1)^{\ell}(2x)^{n-2\ell}}{\ell!\;(n-2\ell)!}=(2x)^{n}{{% }_{2}F_{0}}\left({-\tfrac{1}{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}\atop-};-\frac{1}{% x^{2}}\right).}}
    0 references
    DLMF id
    DLMF:18.5.E13
    0 references
    Symbols used
    Q11125
    Defining formula
    F q p ⁡ ( a 1 , … , a p ; b 1 , … , b q ; z ) Gauss-hypergeometric-pFq 𝑝 𝑞 subscript 𝑎 1 … subscript 𝑎 𝑝 subscript 𝑏 1 … subscript 𝑏 𝑞 𝑧 {\displaystyle{\displaystyle{{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots% ,a_{p}};\NVar{b_{1},\dots,b_{q}};\NVar{z}\right)}}
    xml-id
    C16.S2.m1adec
    0 references
    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.m2aadec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5ahdec
    0 references
    Q11169
    Defining formula
    ⌊ x ⌋ 𝑥 {\displaystyle{\displaystyle\left\lfloor\NVar{x}\right\rfloor}}
    xml-id
    introduction.Sx4.p1.t1.r17.m4aadec
    0 references
    Q11744
    Defining formula
    ℓ ℓ {\displaystyle{\displaystyle\ell}}
    xml-id
    C18.S1.XMD4.m1edec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1mdec
    0 references
    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1ldec
    0 references
     
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