DLMF:24.8.E8 (Q7495): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q12113 / rank
 
Normal rank
Property / Symbols used: Q12113 / qualifier
 
test:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q12113 / qualifier
 
xml-id: C24.S1.XMD2.m1gdec
Property / Symbols used
 
Property / Symbols used: Q12110 / rank
 
Normal rank
Property / Symbols used: Q12110 / qualifier
 
test:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q12110 / qualifier
 
xml-id: C24.S1.XMD5.m1gdec
Property / Symbols used
 
Property / Symbols used: Q12118 / rank
 
Normal rank
Property / Symbols used: Q12118 / qualifier
 
test:

α 𝛼 {\displaystyle{\displaystyle\alpha}}

\alpha
Property / Symbols used: Q12118 / qualifier
 
xml-id: C24.S8.XMD1.m1dec
Property / Symbols used
 
Property / Symbols used: Q12119 / rank
 
Normal rank
Property / Symbols used: Q12119 / qualifier
 
test:

β 𝛽 {\displaystyle{\displaystyle\beta}}

\beta
Property / Symbols used: Q12119 / qualifier
 
xml-id: C24.S8.XMD2.m1dec

Latest revision as of 00:37, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:24.8.E8
No description defined

    Statements

    test
    B 2 ⁢ n 4 ⁢ n ⁢ ( α n - ( - β ) n ) = α n ⁢ ∑ k = 1 ∞ k 2 ⁢ n - 1 e 2 ⁢ α ⁢ k - 1 - ( - β ) n ⁢ ∑ k = 1 ∞ k 2 ⁢ n - 1 e 2 ⁢ β ⁢ k - 1 , Bernoulli-number-B 2 𝑛 4 𝑛 superscript 𝛼 𝑛 superscript 𝛽 𝑛 superscript 𝛼 𝑛 superscript subscript 𝑘 1 superscript 𝑘 2 𝑛 1 superscript 𝑒 2 𝛼 𝑘 1 superscript 𝛽 𝑛 superscript subscript 𝑘 1 superscript 𝑘 2 𝑛 1 superscript 𝑒 2 𝛽 𝑘 1 {\displaystyle{\displaystyle\frac{B_{2n}}{4n}\left(\alpha^{n}-(-\beta)^{n}% \right)=\alpha^{n}\sum_{k=1}^{\infty}\frac{k^{2n-1}}{e^{2\alpha k}-1}-(-\beta)% ^{n}\sum_{k=1}^{\infty}\frac{k^{2n-1}}{e^{2\beta k}-1},}}
    0 references
    DLMF id
    DLMF:24.8.E8
    0 references
    constraint
    n = 2 , 3 , … 𝑛 2 3 … {\displaystyle{\displaystyle n=2,3,\dots}}
    0 references
    Symbols used
    Q11109
    test
    B n Bernoulli-number-B 𝑛 {\displaystyle{\displaystyle B_{\NVar{n}}}}
    xml-id
    C24.S2.SS1.m1abdec
    0 references
    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2acdec
    0 references
    Q12113
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C24.S1.XMD2.m1gdec
    0 references
    Q12110
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C24.S1.XMD5.m1gdec
    0 references
    Q12118
    test
    α 𝛼 {\displaystyle{\displaystyle\alpha}}
    xml-id
    C24.S8.XMD1.m1dec
    0 references
    Q12119
    test
    β 𝛽 {\displaystyle{\displaystyle\beta}}
    xml-id
    C24.S8.XMD2.m1dec
    0 references
     
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