DLMF:25.6.E13 (Q7650): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q12155 / rank
 
Normal rank
Property / Symbols used: Q12155 / qualifier
 
Defining formula:

c 𝑐 {\displaystyle{\displaystyle c}}

c
Property / Symbols used: Q12155 / qualifier
 
xml-id: C25.S4.XMD1.m1dec

Latest revision as of 13:32, 2 January 2020

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DLMF:25.6.E13
No description defined

    Statements

    Defining formula
    ( - 1 ) k ⁢ ζ ( k ) ⁡ ( - 2 ⁢ n ) = 2 ⁢ ( - 1 ) n ( 2 ⁢ π ) 2 ⁢ n + 1 ⁢ ∑ m = 0 k ∑ r = 0 m ( k m ) ⁢ ( m r ) ⁢ ℑ ⁡ ( c k - m ) ⁢ Γ ( r ) ⁡ ( 2 ⁢ n + 1 ) ⁢ ζ ( m - r ) ⁡ ( 2 ⁢ n + 1 ) , superscript 1 𝑘 Riemann-zeta 𝑘 2 𝑛 2 superscript 1 𝑛 superscript 2 𝜋 2 𝑛 1 superscript subscript 𝑚 0 𝑘 superscript subscript 𝑟 0 𝑚 binomial 𝑘 𝑚 binomial 𝑚 𝑟 superscript 𝑐 𝑘 𝑚 Euler-Gamma 𝑟 2 𝑛 1 Riemann-zeta 𝑚 𝑟 2 𝑛 1 {\displaystyle{\displaystyle(-1)^{k}{\zeta^{(k)}}\left(-2n\right)=\frac{2(-1)^% {n}}{(2\pi)^{2n+1}}\sum_{m=0}^{k}\sum_{r=0}^{m}\genfrac{(}{)}{0.0pt}{}{k}{m}% \genfrac{(}{)}{0.0pt}{}{m}{r}\Im(c^{k-m})\*{\Gamma^{(r)}}\left(2n+1\right){% \zeta^{(m-r)}}\left(2n+1\right),}}
    0 references
    DLMF id
    DLMF:25.6.E13
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
    0 references
    Riemann zeta function
    Defining formula
    ζ ⁡ ( s ) Riemann-zeta 𝑠 {\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}
    xml-id
    C25.S2.E1.m2aldec
    0 references
    Q10754
    Defining formula
    ( m n ) binomial 𝑚 𝑛 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}
    xml-id
    C1.S2.SS1.m1acdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aedec
    0 references
    Q10901
    Defining formula
    ℑ ⁡ absent {\displaystyle{\displaystyle\Im}}
    xml-id
    C1.S9.E2.m2adec
    0 references
    Q12150
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C25.S1.XMD1.m1edec
    0 references
    Q12151
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C25.S1.XMD2.m1dec
    0 references
    Q12148
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C25.S1.XMD3.m1ddec
    0 references
    Q12155
    Defining formula
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C25.S4.XMD1.m1dec
    0 references
     
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