Generirajuća funkcija
uredi
x
e
x
−
1
=
∑
k
=
0
∞
B
k
x
k
k
!
=
1
−
1
2
x
+
1
6
x
2
2
!
−
1
30
x
4
4
!
+
1
42
x
6
6
!
−
1
30
x
8
8
!
+
5
66
x
10
10
!
+
…
{\displaystyle {\frac {x}{e^{x}-1}}=\sum _{k=0}^{\infty }B_{k}\,{\frac {x^{k}}{k!}}=1-{\frac {1}{2}}\,x+{\frac {1}{6}}\,{\frac {x^{2}}{2!}}-{\frac {1}{30}}\,{\frac {x^{4}}{4!}}+{\frac {1}{42}}\,{\frac {x^{6}}{6!}}-{\frac {1}{30}}\,{\frac {x^{8}}{8!}}+{\frac {5}{66}}\,{\frac {x^{10}}{10!}}+\ldots }
za
|
x
|
<
π
{\displaystyle |x|<\pi }
Rekurzivna formula
uredi
B
0
=
1
,
{\displaystyle \displaystyle {B_{0}=1\;,}}
B
n
=
−
1
n
+
1
∑
k
=
1
n
(
n
+
1
k
+
1
)
B
n
−
k
,
n
∈
N
.
{\displaystyle B_{n}={\frac {-1}{n+1}}\sum _{k=1}^{n}{n+1 \choose {k+1}}B_{n-k},\quad n\in \mathbb {N} .}
Ojler-Maklorenova formula , koja se koristi za asimptotska računanja integrala prikazana je pomoću Bernulijevih brojeva:
∑
a
≤
k
<
b
f
(
k
)
=
∫
a
b
f
(
x
)
d
x
+
∑
k
=
1
m
B
k
k
!
(
f
(
k
−
1
)
(
b
)
−
f
(
k
−
1
)
(
a
)
)
+
R
(
f
,
m
)
.
{\displaystyle \sum \limits _{a\leq k<b}f(k)=\int _{a}^{b}f(x)\,dx\ +\sum \limits _{k=1}^{m}{\frac {B_{k}}{k!}}\left(f^{(k-1)}(b)-f^{(k-1)}(a)\right)+R(f,m).}
Bernulijevi brojevi koriste se i prilikom razvoja sledećih funkcija:
x
ctg
x
=
∑
n
=
0
∞
(
−
1
)
n
B
2
n
2
2
n
(
2
n
)
!
x
2
n
,
|
x
|
<
π
{\displaystyle x\;\operatorname {ctg} x=\sum _{n=0}^{\infty }(-1)^{n}B_{2n}{\frac {2^{2n}}{(2n)!}}x^{2n},|x|<\pi }
tg
x
=
∑
n
=
1
∞
|
B
2
n
|
2
2
n
(
2
2
n
−
1
)
(
2
n
)
!
x
2
n
−
1
,
|
x
|
<
π
/
2
{\displaystyle \operatorname {tg} x=\sum _{n=1}^{\infty }|B_{2n}|{\frac {2^{2n}(2^{2n}-1)}{(2n)!}}x^{2n-1},|x|<\pi /2}
Leonard Ojler je našao vezu između Bernulijevih brojeva i Rimanove zeta-funkcije ζ(s ) za parne s = 2k :
B
2
k
=
2
(
−
1
)
k
+
1
ζ
(
2
k
)
(
2
k
)
!
(
2
π
)
2
k
.
{\displaystyle B_{2k}=2(-1)^{k+1}{\frac {\zeta (2k)\;(2k)!}{(2\pi )^{2k}}}.}
Odatle sledi:
B
n
=
−
n
ζ
(
1
−
n
)
{\displaystyle \displaystyle {B_{n}=-n\zeta (1-n)}\;\;}
n . Osim toga Bernulijevi brojevi povezani su i sa sledećim integralom:
∫
0
∞
x
2
n
−
1
d
x
e
2
π
x
−
1
=
1
4
n
|
B
2
n
|
,
n
=
1
,
2
,
…
.
{\displaystyle \int \limits _{0}^{\infty }{\frac {x^{2n-1}dx}{e^{2\pi x}-1}}={\frac {1}{4n}}|B_{2n}|,\quad n=1,2,\dots .}
Literatura
uredi
Abramowitz, Milton; Stegun, Irene A., eds. (1965), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, ISBN 978-0486612720
Bernulijevi brojevi
