Polinomios de Bernoulli En matemáticas , los polinomios de Bernoulli
B
n
(
x
)
{\displaystyle B_{n}(x)}
función generatriz :
t
e
x
t
e
t
− − -->
1
=
∑ ∑ -->
n
=
0
∞ ∞ -->
B
n
(
x
)
t
n
n
!
{\displaystyle {\frac {te^{xt}}{e^{t}-1}}=\sum _{n=0}^{\infty }B_{n}(x){\frac {t^{n}}{n!}}}
Aparecen en el estudio de numerosas funciones especiales , en particular de la función zeta de Riemann y de la función zeta de Hurwitz . Los números de Bernoulli
B
n
{\displaystyle B_{n}}
B
n
=
B
n
(
0
)
{\displaystyle B_{n}=B_{n}(0)}
La identidad
B
p
+
1
(
x
+
1
)
− − -->
B
p
+
1
(
x
)
=
(
p
+
1
)
x
p
{\displaystyle B_{p+1}(x+1)-B_{p+1}(x)=(p+1)x^{p}\,}
∑ ∑ -->
m
=
0
n
i
p
=
1
p
+
2
p
+
⋯ ⋯ -->
+
n
p
=
B
p
+
1
(
n
+
1
)
− − -->
B
p
+
1
(
0
)
p
+
1
{\displaystyle \sum _{m=0}^{n}{i^{p}}=1^{p}+2^{p}+\cdots +n^{p}={\frac {B_{p+1}(n+1)-B_{p+1}(0)}{p+1}}}
B
p
(
x
)
=
∑ ∑ -->
m
=
0
p
(
− − -->
1
)
m
(
p
m
)
B
m
⋅ ⋅ -->
x
p
− − -->
m
{\displaystyle B_{p}(x)=\sum _{m=0}^{p}(-1)^{m}{p \choose m}B_{m}\cdot x^{p-m}}
Los primeros Polinomios de Bernoulli son:
B
0
(
x
)
=
1
{\displaystyle B_{0}(x)=1\,}
B
1
(
x
)
=
x
− − -->
1
2
{\displaystyle B_{1}(x)=x-{\frac {1}{2}}\,}
B
2
(
x
)
=
x
2
− − -->
x
+
1
6
{\displaystyle B_{2}(x)=x^{2}-x+{\frac {1}{6}}\,}
B
3
(
x
)
=
x
3
− − -->
3
2
x
2
+
1
2
x
{\displaystyle B_{3}(x)=x^{3}-{\frac {3}{2}}x^{2}+{\frac {1}{2}}x\,}
B
4
(
x
)
=
x
4
− − -->
2
x
3
+
x
2
− − -->
1
30
{\displaystyle B_{4}(x)=x^{4}-2x^{3}+x^{2}-{\frac {1}{30}}\,}
B
5
(
x
)
=
x
5
− − -->
5
2
x
4
+
5
3
x
3
− − -->
1
6
x
{\displaystyle B_{5}(x)=x^{5}-{\frac {5}{2}}x^{4}+{\frac {5}{3}}x^{3}-{\frac {1}{6}}x\,}
B
6
(
x
)
=
x
6
− − -->
3
x
5
+
5
2
x
4
− − -->
1
2
x
2
+
1
42
{\displaystyle B_{6}(x)=x^{6}-3x^{5}+{\frac {5}{2}}x^{4}-{\frac {1}{2}}x^{2}+{\frac {1}{42}}\,}
B
7
(
x
)
=
x
7
− − -->
7
2
x
6
+
7
2
x
5
− − -->
7
6
x
3
+
1
6
x
{\displaystyle B_{7}(x)=x^{7}-{\frac {7}{2}}x^{6}+{\frac {7}{2}}x^{5}-{\frac {7}{6}}x^{3}+{\frac {1}{6}}x\,}
B
8
(
x
)
=
x
8
− − -->
4
x
7
+
14
3
x
6
− − -->
7
3
x
4
+
2
3
x
2
− − -->
1
30
{\displaystyle B_{8}(x)=x^{8}-4x^{7}+{\frac {14}{3}}x^{6}-{\frac {7}{3}}x^{4}+{\frac {2}{3}}x^{2}-{\frac {1}{30}}\,}
B
9
(
x
)
=
x
9
− − -->
9
2
x
8
+
6
x
7
− − -->
21
5
x
5
+
2
x
3
− − -->
3
10
x
{\displaystyle B_{9}(x)=x^{9}-{\frac {9}{2}}x^{8}+6x^{7}-{\frac {21}{5}}x^{5}+2x^{3}-{\frac {3}{10}}x\,}
Recurrencia Integral
En [ 1] [ 2]
B
m
(
x
)
=
m
∫ ∫ -->
0
x
B
m
− − -->
1
(
t
)
d
t
− − -->
m
∫ ∫ -->
0
1
∫ ∫ -->
0
t
B
m
− − -->
1
(
s
)
d
s
d
t
.
{\displaystyle B_{m}(x)=m\int _{0}^{x}B_{m-1}(t)dt-m\int _{0}^{1}\int _{0}^{t}B_{m-1}(s)dsdt.}
Referencias
Enlaces externos
