DLMF:24.4.E24 (Q7437): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11288 / rank
 
Normal rank
Property / Symbols used: Q11288 / qualifier
 
Defining formula:

B n ⁡ ( x ) Bernoulli-polynomial-B 𝑛 𝑥 {\displaystyle{\displaystyle B_{\NVar{n}}\left(\NVar{x}\right)}}

\BernoullipolyB{\NVar{n}}@{\NVar{x}}
Property / Symbols used: Q11288 / qualifier
 
xml-id: C24.S2.SS1.m2akdec
Property / Symbols used
 
Property / Symbols used: Q10754 / rank
 
Normal rank
Property / Symbols used: Q10754 / qualifier
 
Defining formula:

( m n ) binomial 𝑚 𝑛 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}

\binom{\NVar{m}}{\NVar{n}}
Property / Symbols used: Q10754 / qualifier
 
xml-id: C1.S2.SS1.m1agdec
Property / Symbols used
 
Property / Symbols used: the ratio of the circumference of a circle to its diameter / rank
 
Normal rank
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
Defining formula:

π {\displaystyle{\displaystyle\pi}}

\cpi
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
xml-id: C3.S12.E1.m2adec
Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
Defining formula:

e {\displaystyle{\displaystyle\mathrm{e}}}

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2adec
Property / Symbols used
 
Property / Symbols used: imaginary unit / rank
 
Normal rank
Property / Symbols used: imaginary unit / qualifier
 
Defining formula:

i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}

\iunit
Property / Symbols used: imaginary unit / qualifier
 
xml-id: C1.S9.E1.m2adec
Property / Symbols used
 
Property / Symbols used: Q12116 / rank
 
Normal rank
Property / Symbols used: Q12116 / qualifier
 
Defining formula:

j 𝑗 {\displaystyle{\displaystyle j}}

j
Property / Symbols used: Q12116 / qualifier
 
xml-id: C24.S1.XMD1.m1adec
Property / Symbols used
 
Property / Symbols used: Q12113 / rank
 
Normal rank
Property / Symbols used: Q12113 / qualifier
 
Defining formula:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q12113 / qualifier
 
xml-id: C24.S1.XMD2.m1ndec
Property / Symbols used
 
Property / Symbols used: Q12115 / rank
 
Normal rank
Property / Symbols used: Q12115 / qualifier
 
Defining formula:

m 𝑚 {\displaystyle{\displaystyle m}}

m
Property / Symbols used: Q12115 / qualifier
 
xml-id: C24.S1.XMD4.m1hdec
Property / Symbols used
 
Property / Symbols used: Q12110 / rank
 
Normal rank
Property / Symbols used: Q12110 / qualifier
 
Defining formula:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q12110 / qualifier
 
xml-id: C24.S1.XMD5.m1wdec
Property / Symbols used
 
Property / Symbols used: Q12112 / rank
 
Normal rank
Property / Symbols used: Q12112 / qualifier
 
Defining formula:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q12112 / qualifier
 
xml-id: C24.S1.XMD7.m1odec

Latest revision as of 01:28, 2 January 2020

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DLMF:24.4.E24
No description defined

    Statements

    Defining formula
    B n ⁡ ( m ⁢ x ) = m n ⁢ B n ⁡ ( x ) + n ⁢ ∑ k = 1 n ∑ j = 0 k - 1 ( - 1 ) j ⁢ ( n k ) ⁢ ( ∑ r = 1 m - 1 e 2 ⁢ π ⁢ i ⁢ ( k - j ) ⁢ r / m ( 1 - e 2 ⁢ π ⁢ i ⁢ r / m ) n ) ⁢ ( j + m ⁢ x ) n - 1 , Bernoulli-polynomial-B 𝑛 𝑚 𝑥 superscript 𝑚 𝑛 Bernoulli-polynomial-B 𝑛 𝑥 𝑛 superscript subscript 𝑘 1 𝑛 superscript subscript 𝑗 0 𝑘 1 superscript 1 𝑗 binomial 𝑛 𝑘 superscript subscript 𝑟 1 𝑚 1 superscript 𝑒 2 𝜋 𝑖 𝑘 𝑗 𝑟 𝑚 superscript 1 superscript 𝑒 2 𝜋 𝑖 𝑟 𝑚 𝑛 superscript 𝑗 𝑚 𝑥 𝑛 1 {\displaystyle{\displaystyle B_{n}\left(mx\right)=m^{n}B_{n}\left(x\right)+n% \sum_{k=1}^{n}\sum_{j=0}^{k-1}(-1)^{j}{n\choose k}\*\left(\sum_{r=1}^{m-1}% \frac{e^{2\pi i(k-j)r/m}}{(1-e^{2\pi ir/m})^{n}}\right)(j+mx)^{n-1},}}
    0 references
    DLMF id
    DLMF:24.4.E24
    0 references
    constraint
    m = 2 , 3 , … 𝑚 2 3 … {\displaystyle{\displaystyle m=2,3,\dots}}
    0 references
    Symbols used
    Q11288
    Defining formula
    B n ⁡ ( x ) Bernoulli-polynomial-B 𝑛 𝑥 {\displaystyle{\displaystyle B_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C24.S2.SS1.m2akdec
    0 references
    Q10754
    Defining formula
    ( m n ) binomial 𝑚 𝑛 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}
    xml-id
    C1.S2.SS1.m1agdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
    0 references
    Q12116
    Defining formula
    j 𝑗 {\displaystyle{\displaystyle j}}
    xml-id
    C24.S1.XMD1.m1adec
    0 references
    Q12113
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C24.S1.XMD2.m1ndec
    0 references
    Q12115
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C24.S1.XMD4.m1hdec
    0 references
    Q12110
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C24.S1.XMD5.m1wdec
    0 references
    Q12112
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C24.S1.XMD7.m1odec
    0 references
     
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