Bernulli skaitļi sākotnēji parādījās Jākoba Bernulli pēcnāves publikācijā 1713. gadā, kā arī neatkarīgi parādījās Seki Kowa pēcnāves publikācijā 1712. gadā. Bernulli pētīja summas kāpinātiem naturāliem skaitļiem (summas skaitļu kvadrātiem, kubiem, u.t.t.). Definējam
S
p
(
n
)
=
∑
k
=
1
n
−
1
k
p
{\displaystyle S_{p}(n)=\sum _{k=1}^{n-1}k^{p}}
.
Pirmās vērtības:
S
0
(
n
)
=
n
S
1
(
n
)
=
1
2
n
2
−
1
2
n
S
2
(
n
)
=
1
3
n
3
−
1
2
n
2
+
1
6
n
S
3
(
n
)
=
1
4
n
4
−
1
2
n
3
+
1
4
n
2
S
4
(
n
)
=
1
5
n
5
−
1
2
n
4
+
1
3
n
3
−
1
30
n
S
5
(
n
)
=
1
6
n
6
−
1
2
n
5
+
5
12
n
4
−
1
12
n
2
S
6
(
n
)
=
1
7
n
7
−
1
2
n
6
+
1
2
n
5
−
1
6
n
3
+
1
42
n
S
7
(
n
)
=
1
8
n
8
−
1
2
n
7
+
7
12
n
6
−
7
24
n
4
+
1
12
n
2
{\displaystyle {\begin{aligned}&S_{0}(n)=n\\&S_{1}(n)={\frac {1}{2}}n^{2}-{\frac {1}{2}}n\\&S_{2}(n)={\frac {1}{3}}n^{3}-{\frac {1}{2}}n^{2}+{\frac {1}{6}}n\\&S_{3}(n)={\frac {1}{4}}n^{4}-{\frac {1}{2}}n^{3}+{\frac {1}{4}}n^{2}\\&S_{4}(n)={\frac {1}{5}}n^{5}-{\frac {1}{2}}n^{4}+{\frac {1}{3}}n^{3}-{\frac {1}{30}}n\\&S_{5}(n)={\frac {1}{6}}n^{6}-{\frac {1}{2}}n^{5}+{\frac {5}{12}}n^{4}-{\frac {1}{12}}n^{2}\\&S_{6}(n)={\frac {1}{7}}n^{7}-{\frac {1}{2}}n^{6}+{\frac {1}{2}}n^{5}-{\frac {1}{6}}n^{3}+{\frac {1}{42}}n\\&S_{7}(n)={\frac {1}{8}}n^{8}-{\frac {1}{2}}n^{7}+{\frac {7}{12}}n^{6}-{\frac {7}{24}}n^{4}+{\frac {1}{12}}n^{2}\end{aligned}}}
Šīs summas Bernulli pārformulēja, ievērojot to saistību ar kombinācijām . No summas izvelkot koeficientu
1
p
+
1
{\displaystyle {\frac {1}{p+1}}}
un no koeficientiem izvelkot noteiktas kombinācijas iegūst konstantus skaitļus, kurus dēvē par Bernulli skaitļiem.[ 1] Pirmie Bernulli skaitļi virknē:
B
0
=
1
,
B
1
=
−
1
2
,
B
2
=
1
6
,
B
3
=
0
,
B
4
=
−
1
30
,
B
5
=
0
,
B
6
=
1
42
,
B
7
=
0
,
B
8
=
−
1
30
,
B
9
=
0
,
B
10
=
5
66
,
B
11
=
0
{\displaystyle {\begin{aligned}B_{0}=1,\ \ B_{1}=-{\frac {1}{2}},\ \ B_{2}={\frac {1}{6}},\ \ B_{3}=0,\ \ B_{4}=-{\frac {1}{30}},\ \ B_{5}=0,\\B_{6}={\frac {1}{42}},\ \ B_{7}=0,\ \ B_{8}=-{\frac {1}{30}},\ \ B_{9}=0,\ \ B_{10}={\frac {5}{66}},\ \ B_{11}=0\end{aligned}}}
Vispārīgi šo summu var pierakstīt kā
S
p
(
n
)
=
1
p
+
1
∑
k
=
0
p
(
p
+
1
k
)
⋅
B
k
⋅
n
p
+
1
−
k
{\displaystyle S_{p}(n)={\frac {1}{p+1}}\sum _{k=0}^{p}{\binom {p+1}{k}}\cdot B_{k}\cdot n^{p+1-k}}
Bernulli skaitļus var ieviest dažādos veidos:
Bernulli skaitļiem izpildās formulas:
∑
k
=
0
p
(
p
+
1
k
)
B
k
−
=
δ
p
,
0
∑
k
=
0
p
(
p
+
1
k
)
B
k
+
=
m
+
1
{\displaystyle {\begin{aligned}&\sum _{k=0}^{p}{\binom {p+1}{k}}B_{k}^{-}=\delta _{p,0}\\&\sum _{k=0}^{p}{\binom {p+1}{k}}B_{k}^{+}=m+1\end{aligned}}}
kur
m
=
0
,
1
,
2
,
.
.
.
{\displaystyle m=0,1,2,...}
un
δ
{\displaystyle \delta }
ir Kronekera delta . Izsakot
B
m
∓
{\displaystyle B_{m}^{\mp }}
iegūst rekusīvās formulas:
B
p
−
=
δ
p
,
0
−
∑
k
=
0
p
−
1
(
p
k
)
⋅
B
k
−
m
−
k
+
1
B
p
+
=
1
−
∑
k
=
0
p
−
1
(
p
k
)
⋅
B
k
+
m
−
k
+
1
{\displaystyle {\begin{aligned}&B_{p}^{-}=\delta _{p,0}-\sum _{k=0}^{p-1}{\binom {p}{k}}\cdot {\frac {B_{k}^{-}}{m-k+1}}\\&B_{p}^{+}=1-\sum _{k=0}^{p-1}{\binom {p}{k}}\cdot {\frac {B_{k}^{+}}{m-k+1}}\end{aligned}}}
B
p
−
=
∑
k
=
0
p
1
k
+
1
∑
j
=
0
k
(
k
j
)
(
−
1
)
j
j
p
B
p
+
=
∑
k
=
0
p
1
k
+
1
∑
j
=
0
k
(
k
j
)
(
−
1
)
j
(
j
+
1
)
m
{\displaystyle {\begin{aligned}&B_{p}^{-}=\sum _{k=0}^{p}{\frac {1}{k+1}}\sum _{j=0}^{k}{\binom {k}{j}}(-1)^{j}j^{p}\\&B_{p}^{+}=\sum _{k=0}^{p}{\frac {1}{k+1}}\sum _{j=0}^{k}{\binom {k}{j}}(-1)^{j}(j+1)^{m}\end{aligned}}}