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Statements
ζ
(
s
,
a
)
=
1
a
s
+
1
(
1
+
a
)
s
(
1
2
+
1
+
a
s
-
1
)
+
∑
k
=
1
n
(
s
+
2
k
-
2
2
k
-
1
)
B
2
k
2
k
1
(
1
+
a
)
s
+
2
k
-
1
-
(
s
+
2
n
2
n
+
1
)
∫
1
∞
B
~
2
n
+
1
(
x
)
(
x
+
a
)
s
+
2
n
+
1
d
x
,
Hurwitz-zeta
𝑠
𝑎
1
superscript
𝑎
𝑠
1
superscript
1
𝑎
𝑠
1
2
1
𝑎
𝑠
1
superscript
subscript
𝑘
1
𝑛
binomial
𝑠
2
𝑘
2
2
𝑘
1
Bernoulli-number-B
2
𝑘
2
𝑘
1
superscript
1
𝑎
𝑠
2
𝑘
1
binomial
𝑠
2
𝑛
2
𝑛
1
superscript
subscript
1
periodic-Bernoulli-polynomial-B
2
𝑛
1
𝑥
superscript
𝑥
𝑎
𝑠
2
𝑛
1
𝑥
{\displaystyle{\displaystyle\zeta\left(s,a\right)=\frac{1}{a^{s}}+\frac{1}{(1+%
a)^{s}}\left(\frac{1}{2}+\frac{1+a}{s-1}\right)+\sum_{k=1}^{n}\genfrac{(}{)}{0%
.0pt}{}{s+2k-2}{2k-1}\frac{B_{2k}}{2k}\frac{1}{(1+a)^{s+2k-1}}-\genfrac{(}{)}{%
0.0pt}{}{s+2n}{2n+1}\int_{1}^{\infty}\frac{\widetilde{B}_{2n+1}\left(x\right)}%
{(x+a)^{s+2n+1}}\mathrm{d}x,}}
ℜ
s
>
-
2
n
𝑠
2
𝑛
{\displaystyle{\displaystyle\Re s>-2n}}
n
=
1
,
2
,
3
,
…
𝑛
1
2
3
…
{\displaystyle{\displaystyle n=1,2,3,\dots}}
ℜ
s
>
-
2
n
𝑠
2
𝑛
{\displaystyle{\displaystyle\Re s>-2n}}
B
n
Bernoulli-number-B
𝑛
{\displaystyle{\displaystyle B_{\NVar{n}}}}
ζ
(
s
,
a
)
Hurwitz-zeta
𝑠
𝑎
{\displaystyle{\displaystyle\zeta\left(\NVar{s},\NVar{a}\right)}}
(
m
n
)
binomial
𝑚
𝑛
{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
∫
{\displaystyle{\displaystyle\int}}
B
~
n
(
x
)
periodic-Bernoulli-polynomial-B
𝑛
𝑥
{\displaystyle{\displaystyle\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
k
𝑘
{\displaystyle{\displaystyle k}}
n
𝑛
{\displaystyle{\displaystyle n}}
x
𝑥
{\displaystyle{\displaystyle x}}
a
𝑎
{\displaystyle{\displaystyle a}}
s
𝑠
{\displaystyle{\displaystyle s}}
