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Statements
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},}}
m
=
2
,
3
,
…
𝑚
2
3
…
{\displaystyle{\displaystyle m=2,3,\dots}}
B
n
(
x
)
Bernoulli-polynomial-B
𝑛
𝑥
{\displaystyle{\displaystyle B_{\NVar{n}}\left(\NVar{x}\right)}}
(
m
n
)
binomial
𝑚
𝑛
{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}
π
{\displaystyle{\displaystyle\pi}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
i
imaginary-unit
{\displaystyle{\displaystyle\mathrm{i}}}
j
𝑗
{\displaystyle{\displaystyle j}}
k
𝑘
{\displaystyle{\displaystyle k}}
m
𝑚
{\displaystyle{\displaystyle m}}
n
𝑛
{\displaystyle{\displaystyle n}}
x
𝑥
{\displaystyle{\displaystyle x}}
Identifiers
