DLMF:26.14.E4 (Q7955): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: exponential function / rank
 
Normal rank
Property / Symbols used: exponential function / qualifier
 
Defining formula:

exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}

\exp@@{\NVar{z}}
Property / Symbols used: exponential function / qualifier
 
xml-id: C4.S2.E19.m2adec
Property / Symbols used
 
Property / Symbols used: Q10755 / rank
 
Normal rank
Property / Symbols used: Q10755 / qualifier
 
Defining formula:

! {\displaystyle{\displaystyle!}}

!
Property / Symbols used: Q10755 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r15.m5adec
Property / Symbols used
 
Property / Symbols used: Q12173 / rank
 
Normal rank
Property / Symbols used: Q12173 / qualifier
 
Defining formula:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q12173 / qualifier
 
xml-id: C26.S1.XMD1.m1dec
Property / Symbols used
 
Property / Symbols used: Q12174 / rank
 
Normal rank
Property / Symbols used: Q12174 / qualifier
 
Defining formula:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q12174 / qualifier
 
xml-id: C26.S1.XMD4.m1bdec
Property / Symbols used
 
Property / Symbols used: Q12172 / rank
 
Normal rank
Property / Symbols used: Q12172 / qualifier
 
Defining formula:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q12172 / qualifier
 
xml-id: C26.S1.XMD6.m1ddec

Latest revision as of 14:08, 2 January 2020

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DLMF:26.14.E4
No description defined

    Statements

    Defining formula
    ∑ n , k = 0 ∞ ⟨ n k ⟩ ⁢ x k ⁢ t n n ! = 1 - x exp ⁡ ( ( x - 1 ) ⁢ t ) - x , superscript subscript 𝑛 𝑘 0 Eulerian-number 𝑛 𝑘 superscript 𝑥 𝑘 superscript 𝑡 𝑛 𝑛 1 𝑥 𝑥 1 𝑡 𝑥 {\displaystyle{\displaystyle\sum_{n,k=0}^{\infty}\genfrac{<}{>}{0.0pt}{}{n}{k}% x^{k}\,\frac{t^{n}}{n!}=\frac{1-x}{\exp((x-1)t)-x},}}
    0 references
    DLMF id
    DLMF:26.14.E4
    0 references
    constraint
    | x | < 1 , | t | < 1 formulae-sequence 𝑥 1 𝑡 1 {\displaystyle{\displaystyle|x|<1,|t|<1}}
    0 references
    Symbols used
    Q12217
    Defining formula
    ⟨ n k ⟩ Eulerian-number 𝑛 𝑘 {\displaystyle{\displaystyle\genfrac{<}{>}{0.0pt}{}{\NVar{n}}{\NVar{k}}}}
    xml-id
    C26.S14.SS1.p3.m2aadec
    0 references
    exponential function
    Defining formula
    exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}
    xml-id
    C4.S2.E19.m2adec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
    0 references
    Q12173
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C26.S1.XMD1.m1dec
    0 references
    Q12174
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C26.S1.XMD4.m1bdec
    0 references
    Q12172
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C26.S1.XMD6.m1ddec
    0 references
     
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