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F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
( R e d i r e c t e d f r o m B e r n o u l l i p o l y n o m i a l ) Bernoulli polynomials
In mathematics , the Bernoulli polynomials , named after Jacob Bernoulli , combine the Bernoulli numbers and binomial coefficients . They are used for series expansion of functions , and with the Euler–MacLaurin formula .
These polynomials occur in the study of many special functions and, in particular, the Riemann zeta function and the Hurwitz zeta function . They are an Appell sequence (i.e. a Sheffer sequence for the ordinary derivative operator). For the Bernoulli polynomials, the number of crossings of the x unit interval does not go up with the degree . In the limit of large degree, they approach, when appropriately scaled, the sine and cosine functions .
A similar set of polynomials, based on a generating function, is the family of Euler polynomials .
Representations [ edit ]
The Bernoulli polynomials B n generating function . They also admit a variety of derived representations.
Generating functions [ edit ]
The generating function for the Bernoulli polynomials is
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!}}.}
2
e
x t
e
t
+
1
=
∑
n =
0
∞
E
n
(
x )
t
n
n !
.
{\displaystyle {\frac {2e^{xt}}{e^{t}+1}}=\sum _{n=0}^{\infty }E_{n}(x ){\frac {t^{n}}{n!}}.}
Explicit formula [ edit ]
B
n
(
x )
=
∑
k =
0
n
(
n k
)
B
n −
k
x
k
,
{\displaystyle B_{n}(x )=\sum _{k=0}^{n}{n \choose k}B_{n-k}x^{k},}
E
m
(
x )
=
∑
k =
0
m
(
m k
)
E
k
2
k
(
x −
1 2
)
m −
k
.
{\displaystyle E_{m}(x )=\sum _{k=0}^{m}{m \choose k}{\frac {E_{k}}{2^{k}}}\left(x-{\tfrac {1}{2}}\right)^{m-k}.}
n B k Bernoulli numbers , and E k Euler numbers .
Representation by a differential operator [ edit ]
The Bernoulli polynomials are also given by
B
n
(
x )
=
D
e
D
−
1
x
n
{\displaystyle B_{n}(x )={\frac {D}{e^{D}-1}}x^{n}}
D d dx x formal power series . It follows that
∫
a
x
B
n
(
u )
d u =
B
n +
1
(
x )
−
B
n +
1
(
a )
n +
1
.
{\displaystyle \int _{a}^{x}B_{n}(u )\,du={\frac {B_{n+1}(x )-B_{n+1}(a )}{n+1}}.}
§ Integrals below. By the same token, the Euler polynomials are given by
E
n
(
x )
=
2
e
D
+
1
x
n
.
{\displaystyle E_{n}(x )={\frac {2}{e^{D}+1}}x^{n}.}
Representation by an integral operator [ edit ]
The Bernoulli polynomials are also the unique polynomials determined by
∫
x
x +
1
B
n
(
u )
d u =
x
n
.
{\displaystyle \int _{x}^{x+1}B_{n}(u )\,du=x^{n}.}
The integral transform
(
T f )
(
x )
=
∫
x
x +
1
f (
u )
d u
{\displaystyle (Tf )(x )=\int _{x}^{x+1}f(u )\,du}
f
(
T f )
(
x )
=
e
D
−
1
D
f (
x )
=
∑
n =
0
∞
D
n
(
n +
1 )
!
f (
x )
=
f (
x )
+
f ′
(
x )
2
+
f ″
(
x )
6
+
f ‴
(
x )
24
+
⋯
.
{\displaystyle {\begin{aligned}(Tf )(x )={e^{D}-1 \over D}f(x )&{}=\sum _{n=0}^{\infty }{D^{n} \over (n+1)!}f(x )\\&{}=f(x )+{f'(x ) \over 2}+{f''(x ) \over 6}+{f'''(x ) \over 24}+\cdots .\end{aligned}}}
inversion formulae below .
Integral Recurrence [ edit ]
In,[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.}
Another explicit formula [ edit ]
An explicit formula for the Bernoulli polynomials is given by
B
n
(
x )
=
∑
k =
0
n
[
1
k +
1
∑
ℓ
=
0
k
(
−
1
)
ℓ
(
k ℓ
)
(
x +
ℓ
)
n
]
.
{\displaystyle B_{n}(x )=\sum _{k=0}^{n}{\biggl [}{\frac {1}{k+1}}\sum _{\ell =0}^{k}(-1)^{\ell }{k \choose \ell }(x+\ell )^{n}{\biggr ]}.}
That is similar to the series expression for the Hurwitz zeta function in the complex plane. Indeed, there is the relationship
B
n
(
x )
=
−
n ζ
(
1 −
n ,
x )
{\displaystyle B_{n}(x )=-n\zeta (1-n,\,x)}
ζ
(
s ,
q )
{\displaystyle \zeta (s,\,q)}
Hurwitz zeta function . The latter generalizes the Bernoulli polynomials, allowing for non-integer values of n
The inner sum may be understood to be the n th forward difference of
x
m
,
{\displaystyle x^{m},}
Δ
n
x
m
=
∑
k =
0
n
(
−
1
)
n −
k
(
n k
)
(
x +
k
)
m
{\displaystyle \Delta ^{n}x^{m}=\sum _{k=0}^{n}(-1)^{n-k}{n \choose k}(x+k)^{m}}
Δ
{\displaystyle \Delta }
forward difference operator . Thus, one may write
B
n
(
x )
=
∑
k =
0
n
(
−
1
)
k
k +
1
Δ
k
x
n
.
{\displaystyle B_{n}(x )=\sum _{k=0}^{n}{\frac {(-1)^{k}}{k+1}}\Delta ^{k}x^{n}.}
This formula may be derived from an identity appearing above as follows. Since the forward difference operator Δ equals
Δ
=
e
D
−
1
{\displaystyle \Delta =e^{D}-1}
D x Mercator series ,
D
e
D
−
1
=
log
(
Δ
+
1 )
Δ
=
∑
n =
0
∞
(
−
Δ
)
n
n +
1
.
{\displaystyle {\frac {D}{e^{D}-1}}={\frac {\log(\Delta +1)}{\Delta }}=\sum _{n=0}^{\infty }{\frac {(-\Delta )^{n}}{n+1}}.}
As long as this operates on an m
x
m
,
{\displaystyle x^{m},}
n 0 only up to m
An integral representation for the Bernoulli polynomials is given by the Nörlund–Rice integral , which follows from the expression as a finite difference.
An explicit formula for the Euler polynomials is given by
E
n
(
x )
=
∑
k =
0
n
[
1
2
k
∑
ℓ
=
0
n
(
−
1
)
ℓ
(
k ℓ
)
(
x +
ℓ
)
n
]
.
{\displaystyle E_{n}(x )=\sum _{k=0}^{n}\left[{\frac {1}{2^{k}}}\sum _{\ell =0}^{n}(-1)^{\ell }{k \choose \ell }(x+\ell )^{n}\right].}
The above follows analogously, using the fact that
2
e
D
+
1
=
1
1 +
1 2
Δ
=
∑
n =
0
∞
(
−
1 2
Δ
)
n
.
{\displaystyle {\frac {2}{e^{D}+1}}={\frac {1}{1+{\tfrac {1}{2}}\Delta }}=\sum _{n=0}^{\infty }{\bigl (}{-{\tfrac {1}{2}}}\Delta {\bigr )}^{n}.}
Sums of p [ edit ]
Using either the above integral representation of
x
n
{\displaystyle x^{n}}
identity
B
n
(
x +
1 )
−
B
n
(
x )
=
n
x
n −
1
{\displaystyle B_{n}(x+1)-B_{n}(x )=nx^{n-1}}
∑
k =
0
x
k
p
=
∫
0
x +
1
B
p
(
t )
d t =
B
p +
1
(
x +
1 )
−
B
p +
1
p +
1
{\displaystyle \sum _{k=0}^{x}k^{p}=\int _{0}^{x+1}B_{p}(t )\,dt={\frac {B_{p+1}(x+1)-B_{p+1}}{p+1}}}
0 = 1 ).
The Bernoulli and Euler numbers [ edit ]
The Bernoulli numbers are given by
B
n
=
B
n
(
0
)
.
{\textstyle B_{n}=B_{n}(0).}
This definition gives
ζ
(
−
n )
=
(
−
1
)
n
n +
1
B
n +
1
{\textstyle \zeta (-n)={\frac {(-1)^{n}}{n+1}}B_{n+1}}
n =
0
,
1 ,
2 ,
…
.
{\textstyle n=0,\,1,\,2,\,\ldots .}
An alternate convention defines the Bernoulli numbers as
B
n
=
B
n
(
1 )
.
{\textstyle B_{n}=B_{n}(1 ).}
The two conventions differ only when
n =
1 ,
{\displaystyle n=1,}
B
1
(
1 )
=
−
B
1
(
0
)
=
1 2
.
{\displaystyle B_{1}(1 )=-B_{1}(0)={\tfrac {1}{2}}.}
The Euler numbers are given by
E
n
=
2
n
E
n
(
1 2
)
.
{\displaystyle E_{n}=2^{n}E_{n}{\bigl (}{\tfrac {1}{2}}{\bigr )}.}
Explicit expressions for low degrees [ edit ]
The first few Bernoulli polynomials are:
B
0
(
x )
=
1 ,
B
4
(
x )
=
x
4
−
2
x
3
+
x
2
−
1 30
,
B
1
(
x )
=
x −
1 2
,
B
5
(
x )
=
x
5
−
5 2
x
4
+
5 3
x
3
−
1 6
x ,
B
2
(
x )
=
x
2
−
x +
1 6
,
B
6
(
x )
=
x
6
−
3
x
5
+
5 2
x
4
−
1 2
x
2
+
1 42
,
B
3
(
x )
=
x
3
−
3 2
x
2
+
1 2
x
|
,
⋮
{\displaystyle {\begin{aligned}B_{0}(x )&=1,&B_{4}(x )&=x^{4}-2x^{3}+x^{2}-{\tfrac {1}{30}},\\[4mu]B_{1}(x )&=x-{\tfrac {1}{2}},&B_{5}(x )&=x^{5}-{\tfrac {5}{2}}x^{4}+{\tfrac {5}{3}}x^{3}-{\tfrac {1}{6}}x,\\[4mu]B_{2}(x )&=x^{2}-x+{\tfrac {1}{6}},&B_{6}(x )&=x^{6}-3x^{5}+{\tfrac {5}{2}}x^{4}-{\tfrac {1}{2}}x^{2}+{\tfrac {1}{42}},\\[-2mu]B_{3}(x )&=x^{3}-{\tfrac {3}{2}}x^{2}+{\tfrac {1}{2}}x{\vphantom {\Big |}},\qquad &&\ \,\,\vdots \end{aligned}}}
The first few Euler polynomials are:
E
0
(
x )
=
1 ,
E
4
(
x )
=
x
4
−
2
x
3
+
x ,
E
1
(
x )
=
x −
1 2
,
E
5
(
x )
=
x
5
−
5 2
x
4
+
5 2
x
2
−
1 2
,
E
2
(
x )
=
x
2
−
x ,
E
6
(
x )
=
x
6
−
3
x
5
+
5
x
3
−
3 x ,
E
3
(
x )
=
x
3
−
3 2
x
2
+
1 4
,
⋮
{\displaystyle {\begin{aligned}E_{0}(x )&=1,&E_{4}(x )&=x^{4}-2x^{3}+x,\\[4mu]E_{1}(x )&=x-{\tfrac {1}{2}},&E_{5}(x )&=x^{5}-{\tfrac {5}{2}}x^{4}+{\tfrac {5}{2}}x^{2}-{\tfrac {1}{2}},\\[4mu]E_{2}(x )&=x^{2}-x,&E_{6}(x )&=x^{6}-3x^{5}+5x^{3}-3x,\\[-1mu]E_{3}(x )&=x^{3}-{\tfrac {3}{2}}x^{2}+{\tfrac {1}{4}},\qquad \ \ &&\ \,\,\vdots \end{aligned}}}
Maximum and minimum [ edit ]
At higher n
B
n
(
x )
{\displaystyle B_{n}(x )}
x =
0
{\displaystyle x=0}
x =
1
{\displaystyle x=1}
B
16
(
0
)
=
B
16
(
1 )
=
{\displaystyle B_{16}(0)=B_{16}(1 )={}}
−
3617
510
≈
−
7.09
,
{\displaystyle -{\tfrac {3617}{510}}\approx -7.09,}
B
16
(
1 2
)
=
{\displaystyle B_{16}{\bigl (}{\tfrac {1}{2}}{\bigr )}={}}
118518239
3342336
≈
7.09.
{\displaystyle {\tfrac {118518239}{3342336}}\approx 7.09.}
Lehmer (1940)[3] M n
B
n
(
x )
{\displaystyle B_{n}(x )}
0 and 1
M
n
<
2 n !
(
2 π
)
n
{\displaystyle M_{n}<{\frac {2n!}{(2\pi )^{n}}}}
n is 2 modulo 4 ,
M
n
=
2 ζ
(
n )
n !
(
2 π
)
n
{\displaystyle M_{n}={\frac {2\zeta (n )\,n!}{(2\pi )^{n}}}}
ζ
(
x )
{\displaystyle \zeta (x )}
Riemann zeta function ), while the minimum (m n
m
n
>
−
2 n !
(
2 π
)
n
{\displaystyle m_{n}>{\frac {-2n!}{(2\pi )^{n}}}}
n in which case
m
n
=
−
2 ζ
(
n )
n !
(
2 π
)
n
.
{\displaystyle m_{n}={\frac {-2\zeta (n )\,n!}{(2\pi )^{n}}}.}
These limits are quite close to the actual maximum and minimum, and Lehmer gives more accurate limits as well.
Differences and derivatives [ edit ]
The Bernoulli and Euler polynomials obey many relations from umbral calculus :
Δ
B
n
(
x )
=
B
n
(
x +
1 )
−
B
n
(
x )
=
n
x
n −
1
,
Δ
E
n
(
x )
=
E
n
(
x +
1 )
−
E
n
(
x )
=
2 (
x
n
−
E
n
(
x )
)
.
{\displaystyle {\begin{aligned}\Delta B_{n}(x )&=B_{n}(x+1)-B_{n}(x )=nx^{n-1},\\[3mu]\Delta E_{n}(x )&=E_{n}(x+1)-E_{n}(x )=2(x^{n}-E_{n}(x )).\end{aligned}}}
Δ is the forward difference operator ). Also,
E
n
(
x +
1 )
+
E
n
(
x )
=
2
x
n
.
{\displaystyle E_{n}(x+1)+E_{n}(x )=2x^{n}.}
polynomial sequences are Appell sequences :
B
n
′
(
x )
=
n
B
n −
1
(
x )
,
E
n
′
(
x )
=
n
E
n −
1
(
x )
.
{\displaystyle {\begin{aligned}B_{n}'(x )&=nB_{n-1}(x ),\\[3mu]E_{n}'(x )&=nE_{n-1}(x ).\end{aligned}}}
Translations [ edit ]
B
n
(
x +
y )
=
∑
k =
0
n
(
n k
)
B
k
(
x )
y
n −
k
E
n
(
x +
y )
=
∑
k =
0
n
(
n k
)
E
k
(
x )
y
n −
k
{\displaystyle {\begin{aligned}B_{n}(x+y)&=\sum _{k=0}^{n}{n \choose k}B_{k}(x )y^{n-k}\\[3mu]E_{n}(x+y)&=\sum _{k=0}^{n}{n \choose k}E_{k}(x )y^{n-k}\end{aligned}}}
Appell sequences . (Hermite polynomials are another example.)
Symmetries [ edit ]
B
n
(
1 −
x )
=
(
−
1
)
n
B
n
(
x )
,
n ≥
0
,
E
n
(
1 −
x )
=
(
−
1
)
n
E
n
(
x )
(
−
1
)
n
B
n
(
−
x )
=
B
n
(
x )
+
n
x
n −
1
(
−
1
)
n
E
n
(
−
x )
=
−
E
n
(
x )
+
2
x
n
B
n
(
1 2
)
=
(
1
2
n −
1
−
1
)
B
n
,
n ≥
0
from the multiplication theorems below.
{\displaystyle {\begin{aligned}B_{n}(1-x)&=\left(-1\right)^{n}B_{n}(x ),&&n\geq 0,\\[3mu]E_{n}(1-x)&=\left(-1\right)^{n}E_{n}(x )\\[1ex]\left(-1\right)^{n}B_{n}(-x)&=B_{n}(x )+nx^{n-1}\\[3mu]\left(-1\right)^{n}E_{n}(-x)&=-E_{n}(x )+2x^{n}\\[1ex]B_{n}{\bigl (}{\tfrac {1}{2}}{\bigr )}&=\left({\frac {1}{2^{n-1}}}-1\right)B_{n},&&n\geq 0{\text{ from the multiplication theorems below.}}\end{aligned}}}
Zhi-Wei Sun and Hao Pan [4] r s t n x y z
r [
s ,
t ;
x ,
y
]
n
+
s [
t ,
r ;
y ,
z
]
n
+
t [
r ,
s ;
z ,
x
]
n
=
0
,
{\displaystyle r[s,t;x,y]_{n}+s[t,r;y,z]_{n}+t[r,s;z,x]_{n}=0,}
[
s ,
t ;
x ,
y
]
n
=
∑
k =
0
n
(
−
1
)
k
(
s k
)
(
t
n −
k
)
B
n −
k
(
x )
B
k
(
y )
.
{\displaystyle [s,t;x,y]_{n}=\sum _{k=0}^{n}(-1)^{k}{s \choose k}{t \choose {n-k}}B_{n-k}(x )B_{k}(y ).}
Fourier series [ edit ]
The Fourier series of the Bernoulli polynomials is also a Dirichlet series , given by the expansion
B
n
(
x )
=
−
n !
(
2 π
i
)
n
∑
k ≠
0
e
2 π
i k x
k
n
=
−
2 n !
∑
k =
1
∞
cos
(
2 k π
x −
n π
2
)
(
2 k π
)
n
.
{\displaystyle B_{n}(x )=-{\frac {n!}{(2\pi i)^{n}}}\sum _{k\not =0}{\frac {e^{2\pi ikx}}{k^{n}}}=-2n!\sum _{k=1}^{\infty }{\frac {\cos \left(2k\pi x-{\frac {n\pi }{2}}\right)}{(2k\pi )^{n}}}.}
n
This is a special case of the analogous form for the Hurwitz zeta function
B
n
(
x )
=
−
Γ
(
n +
1 )
∑
k =
1
∞
exp
(
2 π
i k x )
+
e
i π
n
exp
(
2 π
i k (
1 −
x )
)
(
2 π
i k
)
n
.
{\displaystyle B_{n}(x )=-\Gamma (n+1)\sum _{k=1}^{\infty }{\frac {\exp(2\pi ikx)+e^{i\pi n}\exp(2\pi ik(1-x))}{(2\pi ik)^{n}}}.}
This expansion is valid only for 0 ≤ x when n 0 < x 1 when n
The Fourier series of the Euler polynomials may also be calculated. Defining the functions
C
ν
(
x )
=
∑
k =
0
∞
cos
(
(
2 k +
1 )
π
x )
(
2 k +
1
)
ν
S
ν
(
x )
=
∑
k =
0
∞
sin
(
(
2 k +
1 )
π
x )
(
2 k +
1
)
ν
{\displaystyle {\begin{aligned}C_{\nu }(x )&=\sum _{k=0}^{\infty }{\frac {\cos((2k+1)\pi x)}{(2k+1)^{\nu }}}\\[3mu]S_{\nu }(x )&=\sum _{k=0}^{\infty }{\frac {\sin((2k+1)\pi x)}{(2k+1)^{\nu }}}\end{aligned}}}
ν
>
1
{\displaystyle \nu >1}
C
2 n
(
x )
=
(
−
1
)
n
4 (
2 n −
1 )
!
π
2 n
E
2 n −
1
(
x )
S
2 n +
1
(
x )
=
(
−
1
)
n
4 (
2 n )
!
π
2 n +
1
E
2 n
(
x )
.
{\displaystyle {\begin{aligned}C_{2n}(x )&={\frac {\left(-1\right)^{n}}{4(2n-1)!}}\pi ^{2n}E_{2n-1}(x )\\[1ex]S_{2n+1}(x )&={\frac {\left(-1\right)^{n}}{4(2n)!}}\pi ^{2n+1}E_{2n}(x ).\end{aligned}}}
C
ν
{\displaystyle C_{\nu }}
S
ν
{\displaystyle S_{\nu }}
C
ν
(
x )
=
−
C
ν
(
1 −
x )
S
ν
(
x )
=
S
ν
(
1 −
x )
.
{\displaystyle {\begin{aligned}C_{\nu }(x )&=-C_{\nu }(1-x)\\S_{\nu }(x )&=S_{\nu }(1-x).\end{aligned}}}
They are related to the Legendre chi function
χ
ν
{\displaystyle \chi _{\nu }}
as
C
ν
(
x )
=
Re
χ
ν
(
e
i x
)
S
ν
(
x )
=
Im
χ
ν
(
e
i x
)
.
{\displaystyle {\begin{aligned}C_{\nu }(x )&=\operatorname {Re} \chi _{\nu }(e^{ix})\\S_{\nu }(x )&=\operatorname {Im} \chi _{\nu }(e^{ix}).\end{aligned}}}
Inversion [ edit ]
The Bernoulli and Euler polynomials may be inverted to express the monomial in terms of the polynomials.
Specifically, evidently from the above section on integral operators , it follows that
x
n
=
1
n +
1
∑
k =
0
n
(
n +
1
k
)
B
k
(
x )
{\displaystyle x^{n}={\frac {1}{n+1}}\sum _{k=0}^{n}{n+1 \choose k}B_{k}(x )}
x
n
=
E
n
(
x )
+
1 2
∑
k =
0
n −
1
(
n k
)
E
k
(
x )
.
{\displaystyle x^{n}=E_{n}(x )+{\frac {1}{2}}\sum _{k=0}^{n-1}{n \choose k}E_{k}(x ).}
Relation to falling factorial [ edit ]
The Bernoulli polynomials may be expanded in terms of the falling factorial
(
x
)
k
{\displaystyle (x )_{k}}
as
B
n +
1
(
x )
=
B
n +
1
+
∑
k =
0
n
n +
1
k +
1
{
n
k
}
(
x
)
k +
1
{\displaystyle B_{n+1}(x )=B_{n+1}+\sum _{k=0}^{n}{\frac {n+1}{k+1}}\left\{{\begin{matrix}n\\k\end{matrix}}\right\}(x )_{k+1}}
B
n
=
B
n
(
0
)
{\displaystyle B_{n}=B_{n}(0)}
{
n
k
}
=
S (
n ,
k )
{\displaystyle \left\{{\begin{matrix}n\\k\end{matrix}}\right\}=S(n,k)}
Stirling number of the second kind . The above may be inverted to express the falling factorial in terms of the Bernoulli polynomials:
(
x
)
n +
1
=
∑
k =
0
n
n +
1
k +
1
[
n
k
]
(
B
k +
1
(
x )
−
B
k +
1
)
{\displaystyle (x )_{n+1}=\sum _{k=0}^{n}{\frac {n+1}{k+1}}\left[{\begin{matrix}n\\k\end{matrix}}\right]\left(B_{k+1}(x )-B_{k+1}\right)}
[
n
k
]
=
s (
n ,
k )
{\displaystyle \left[{\begin{matrix}n\\k\end{matrix}}\right]=s(n,k)}
Stirling number of the first kind .
Multiplication theorems [ edit ]
The multiplication theorems were given by Joseph Ludwig Raabe in 1851:
For a natural number m
B
n
(
m x )
=
m
n −
1
∑
k =
0
m −
1
B
n
(
x +
k m
)
{\displaystyle B_{n}(mx )=m^{n-1}\sum _{k=0}^{m-1}B_{n}{\left(x+{\frac {k}{m}}\right)}}
E
n
(
m x )
=
m
n
∑
k =
0
m −
1
(
−
1
)
k
E
n
(
x +
k m
)
for odd
m
E
n
(
m x )
=
−
2
n +
1
m
n
∑
k =
0
m −
1
(
−
1
)
k
B
n +
1
(
x +
k m
)
for even
m
{\displaystyle {\begin{aligned}E_{n}(mx )&=m^{n}\sum _{k=0}^{m-1}\left(-1\right)^{k}E_{n}{\left(x+{\frac {k}{m}}\right)}&{\text{ for odd }}m\\[1ex]E_{n}(mx )&={\frac {-2}{n+1}}m^{n}\sum _{k=0}^{m-1}\left(-1\right)^{k}B_{n+1}{\left(x+{\frac {k}{m}}\right)}&{\text{ for even }}m\end{aligned}}}
Integrals [ edit ]
Two definite integrals relating the Bernoulli and Euler polynomials to the Bernoulli and Euler numbers are:[5]
∫
0
1
B
n
(
t )
B
m
(
t )
d t =
(
−
1
)
n −
1
m ! n !
(
m +
n )
!
B
n +
m
for
m ,
n ≥
1
{\displaystyle \int _{0}^{1}B_{n}(t )B_{m}(t )\,dt=(-1)^{n-1}{\frac {m!\,n!}{(m+n)!}}B_{n+m}\quad {\text{for }}m,n\geq 1}
∫
0
1
E
n
(
t )
E
m
(
t )
d t =
(
−
1
)
n
4 (
2
m +
n +
2
−
1 )
m ! n !
(
m +
n +
2 )
!
B
n +
m +
2
{\displaystyle \int _{0}^{1}E_{n}(t )E_{m}(t )\,dt=(-1)^{n}4(2^{m+n+2}-1){\frac {m!\,n!}{(m+n+2)!}}B_{n+m+2}}
Another integral formula states[6]
∫
0
1
E
n
(
x +
y
)
log
(
tan
π
2
x )
d x =
n !
∑
k =
1
⌊
n +
1
2
⌋
(
−
1
)
k −
1
π
2 k
(
2 −
2
−
2 k
)
ζ
(
2 k +
1 )
y
n +
1 −
2 k
(
n +
1 −
2 k )
!
{\displaystyle \int _{0}^{1}E_{n}\left(x+y\right)\log(\tan {\frac {\pi }{2}}x)\,dx=n!\sum _{k=1}^{\left\lfloor {\frac {n+1}{2}}\right\rfloor }{\frac {(-1)^{k-1}}{\pi ^{2k}}}\left(2-2^{-2k}\right)\zeta (2k+1){\frac {y^{n+1-2k}}{(n+1-2k)!}}}
with the special case for
y =
0
{\displaystyle y=0}
∫
0
1
E
2 n −
1
(
x )
log
(
tan
π
2
x )
d x =
(
−
1
)
n −
1
(
2 n −
1 )
!
π
2 n
(
2 −
2
−
2 n
)
ζ
(
2 n +
1 )
{\displaystyle \int _{0}^{1}E_{2n-1}\left(x\right)\log(\tan {\frac {\pi }{2}}x)\,dx={\frac {(-1)^{n-1}(2n-1)!}{\pi ^{2n}}}\left(2-2^{-2n}\right)\zeta (2n+1)}
∫
0
1
B
2 n −
1
(
x )
log
(
tan
π
2
x )
d x =
(
−
1
)
n −
1
π
2 n
2
2 n −
2
(
2 n −
1 )
!
∑
k =
1
n
(
2
2 k +
1
−
1 )
ζ
(
2 k +
1 )
ζ
(
2 n −
2 k )
{\displaystyle \int _{0}^{1}B_{2n-1}\left(x\right)\log(\tan {\frac {\pi }{2}}x)\,dx={\frac {(-1)^{n-1}}{\pi ^{2n}}}{\frac {2^{2n-2}}{(2n-1)!}}\sum _{k=1}^{n}(2^{2k+1}-1)\zeta (2k+1)\zeta (2n-2k)}
∫
0
1
E
2 n
(
x )
log
(
tan
π
2
x )
d x =
∫
0
1
B
2 n
(
x )
log
(
tan
π
2
x )
d x =
0
{\displaystyle \int _{0}^{1}E_{2n}\left(x\right)\log(\tan {\frac {\pi }{2}}x)\,dx=\int _{0}^{1}B_{2n}\left(x\right)\log(\tan {\frac {\pi }{2}}x)\,dx=0}
∫
0
1
B
2 n −
1
(
x )
cot
(
π
x
)
d x
=
2
(
2 n −
1
)
!
(
−
1
)
n −
1
(
2 π
)
2 n −
1
ζ
(
2 n −
1
)
{\displaystyle \int _{0}^{1}{{{B}_{2n-1}}\left(x\right)\cot \left(\pi x\right)dx}={\frac {2\left(2n-1\right)!}{{{\left(-1\right)}^{n-1}}{{\left(2\pi \right)}^{2n-1}}}}\zeta \left(2n-1\right)}
Periodic Bernoulli polynomials [ edit ]
A periodic Bernoulli polynomial P n x fractional part of the argument x remainder term in the Euler–Maclaurin formula relating sums to integrals. The first polynomial is a sawtooth function .
Strictly these functions are not polynomials at all and more properly should be termed the periodic Bernoulli functions, and P 0 (x Dirac comb .
The following properties are of interest, valid for all
x
{\displaystyle x}
P
k
(
x )
{\displaystyle P_{k}(x )}
k >
1
{\displaystyle k>1}
P
k
′
(
x )
{\displaystyle P_{k}'(x )}
k >
2
{\displaystyle k>2}
P
k
′
(
x )
=
k
P
k −
1
(
x )
{\displaystyle P'_{k}(x )=kP_{k-1}(x )}
k >
2
{\displaystyle k>2}
See also [ edit ]
References [ edit ]
^ Lehmer, D.H. (1940). "On the maxima and minima of Bernoulli polynomials". American Mathematical Monthly 47
^ Zhi-Wei Sun; Hao Pan (2006). "Identities concerning Bernoulli and Euler polynomials". Acta Arithmetica . 125 (1 ): 21–39. arXiv :math/0409035 Bibcode :2006AcAri.125...21S . doi :10.4064/aa125-1-3 . S2CID 10841415 .
^ Takashi Agoh & Karl Dilcher (2011). "Integrals of products of Bernoulli polynomials" . Journal of Mathematical Analysis and Applications . 381 : 10–16. doi :10.1016/j.jmaa.2011.03.061
^ Elaissaoui, Lahoucine & Guennoun, Zine El Abidine (2017). "Evaluation of log-tangent integrals by series involving ζ(2n+1)". Integral Transforms and Special Functions . 28 6 ): 460–475. arXiv :1611.01274 doi :10.1080/10652469.2017.1312366 . S2CID 119132354 .
Apostol, Tom M. (1976), Introduction to analytic number theory , Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3 MR 0434929 , Zbl 0335.10001 (See chapter 12.11)
Dilcher, K. (2010), "Bernoulli and Euler Polynomials" , in Olver, Frank W. J. ; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions ISBN 978-0-521-19225-5 MR 2723248
Cvijović, Djurdje; Klinowski, Jacek (1995). "New formulae for the Bernoulli and Euler polynomials at rational arguments" . Proceedings of the American Mathematical Society 123 (5 ): 1527–1535. doi :10.1090/S0002-9939-1995-1283544-0 JSTOR 2161144 .
Guillera, Jesus; Sondow, Jonathan (2008). "Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent". The Ramanujan Journal . 16 3 ): 247–270. arXiv :math.NT/0506319 doi :10.1007/s11139-007-9102-0 . S2CID 14910435 . (Reviews relationship to the Hurwitz zeta function and Lerch transcendent.)
Hugh L. Montgomery ; Robert C. Vaughan (2007). Multiplicative number theory I. Classical theory . Cambridge tracts in advanced mathematics. Vol. 97. Cambridge: Cambridge Univ. Press. pp. 495–519. ISBN 978-0-521-84903-6
External links [ edit ]
International
National
Other
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Bernoulli_polynomials&oldid=1226021096 " C a t e g o r i e s : ● S p e c i a l f u n c t i o n s ● N u m b e r t h e o r y ● P o l y n o m i a l s H i d d e n c a t e g o r i e s : ● U s e A m e r i c a n E n g l i s h f r o m M a r c h 2 0 1 9 ● A l l W i k i p e d i a a r t i c l e s w r i t t e n i n A m e r i c a n E n g l i s h ● A r t i c l e s w i t h s h o r t d e s c r i p t i o n ● S h o r t d e s c r i p t i o n m a t c h e s W i k i d a t a ● A r t i c l e s w i t h F A S T i d e n t i f i e r s ● A r t i c l e s w i t h B N F i d e n t i f i e r s ● A r t i c l e s w i t h B N F d a t a i d e n t i f i e r s ● A r t i c l e s w i t h G N D i d e n t i f i e r s ● A r t i c l e s w i t h J 9 U i d e n t i f i e r s ● A r t i c l e s w i t h L C C N i d e n t i f i e r s ● A r t i c l e s w i t h S U D O C i d e n t i f i e r s
● T h i s p a g e w a s l a s t e d i t e d o n 2 8 M a y 2 0 2 4 , a t 0 3 : 3 6 ( U T C ) . ● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n . ● P r i v a c y p o l i c y ● A b o u t W i k i p e d i a ● D i s c l a i m e r s ● C o n t a c t W i k i p e d i a ● C o d e o f C o n d u c t ● D e v e l o p e r s ● S t a t i s t i c s ● C o o k i e s t a t e m e n t ● M o b i l e v i e w
The generating function for the Euler polynomials is
for n ≥ 0, where Bk are the Bernoulli numbers, and Ek are the Euler numbers.
where D = d/dx is differentiation with respect to x and the fraction is expanded as a formal power series. It follows that
cf. § Integrals below. By the same token, the Euler polynomials are given by
on polynomials f, simply amounts to
This can be used to produce the inversion formulae below.
![{\displaystyle B_{n}(x)=\sum _{k=0}^{n}{\biggl [}{\frac {1}{k+1}}\sum _{\ell =0}^{k}(-1)^{\ell }{k \choose \ell }(x+\ell )^{n}{\biggr ]}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c40ae0f284287110f93bff7d95eccd3735dc0f0e)
where 
is the Hurwitz zeta function. The latter generalizes the Bernoulli polynomials, allowing for non-integer values ofn.
where 
is the forward difference operator. Thus, one may write
where D is differentiation with respect to x, we have, from the Mercator series,
![{\displaystyle E_{n}(x)=\sum _{k=0}^{n}\left[{\frac {1}{2^{k}}}\sum _{\ell =0}^{n}(-1)^{\ell }{k \choose \ell }(x+\ell )^{n}\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd5be65f7f49900a001efb6a36d48f042f5fcbb1)
(assuming 00 = 1).
![{\displaystyle {\begin{aligned}B_{0}(x)&=1,&B_{4}(x)&=x^{4}-2x^{3}+x^{2}-{\tfrac {1}{30}},\\[4mu]B_{1}(x)&=x-{\tfrac {1}{2}},&B_{5}(x)&=x^{5}-{\tfrac {5}{2}}x^{4}+{\tfrac {5}{3}}x^{3}-{\tfrac {1}{6}}x,\\[4mu]B_{2}(x)&=x^{2}-x+{\tfrac {1}{6}},&B_{6}(x)&=x^{6}-3x^{5}+{\tfrac {5}{2}}x^{4}-{\tfrac {1}{2}}x^{2}+{\tfrac {1}{42}},\\[-2mu]B_{3}(x)&=x^{3}-{\tfrac {3}{2}}x^{2}+{\tfrac {1}{2}}x{\vphantom {\Big |}},\qquad &&\ \,\,\vdots \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba35b7c4288361b2761e9914aba3d9a59aed02af)
![{\displaystyle {\begin{aligned}E_{0}(x)&=1,&E_{4}(x)&=x^{4}-2x^{3}+x,\\[4mu]E_{1}(x)&=x-{\tfrac {1}{2}},&E_{5}(x)&=x^{5}-{\tfrac {5}{2}}x^{4}+{\tfrac {5}{2}}x^{2}-{\tfrac {1}{2}},\\[4mu]E_{2}(x)&=x^{2}-x,&E_{6}(x)&=x^{6}-3x^{5}+5x^{3}-3x,\\[-1mu]E_{3}(x)&=x^{3}-{\tfrac {3}{2}}x^{2}+{\tfrac {1}{4}},\qquad \ \ &&\ \,\,\vdots \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0eabd8f0560b91e8d9f46e88f5aa019c0eabae98)
unless nis2 modulo 4, in which case
(where 
is the Riemann zeta function), while the minimum (mn) obeys
unless n = 0 modulo 4 , in which case
![{\displaystyle {\begin{aligned}\Delta B_{n}(x)&=B_{n}(x+1)-B_{n}(x)=nx^{n-1},\\[3mu]\Delta E_{n}(x)&=E_{n}(x+1)-E_{n}(x)=2(x^{n}-E_{n}(x)).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19173f56a1a26481b718dad8719ade4c8b851f67)
(Δ is the forward difference operator). Also,
These polynomial sequences are Appell sequences:
![{\displaystyle {\begin{aligned}B_{n}'(x)&=nB_{n-1}(x),\\[3mu]E_{n}'(x)&=nE_{n-1}(x).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc5c456f0e163529d714ec585580fbc0012d4ada)
![{\displaystyle {\begin{aligned}B_{n}(x+y)&=\sum _{k=0}^{n}{n \choose k}B_{k}(x)y^{n-k}\\[3mu]E_{n}(x+y)&=\sum _{k=0}^{n}{n \choose k}E_{k}(x)y^{n-k}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc816e88dcb52c1861c1a26cf9e583b7108a922)
These identities are also equivalent to saying that these polynomial sequences are Appell sequences. (Hermite polynomials are another example.)
![{\displaystyle {\begin{aligned}B_{n}(1-x)&=\left(-1\right)^{n}B_{n}(x),&&n\geq 0,\\[3mu]E_{n}(1-x)&=\left(-1\right)^{n}E_{n}(x)\\[1ex]\left(-1\right)^{n}B_{n}(-x)&=B_{n}(x)+nx^{n-1}\\[3mu]\left(-1\right)^{n}E_{n}(-x)&=-E_{n}(x)+2x^{n}\\[1ex]B_{n}{\bigl (}{\tfrac {1}{2}}{\bigr )}&=\left({\frac {1}{2^{n-1}}}-1\right)B_{n},&&n\geq 0{\text{ from the multiplication theorems below.}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7474656bcca9e16cf848e1e953476473cb826482)
Zhi-Wei Sun and Hao Pan [4] established the following surprising symmetry relation: If r + s + t = n and x + y + z = 1, then
![{\displaystyle r[s,t;x,y]_{n}+s[t,r;y,z]_{n}+t[r,s;z,x]_{n}=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/379cf436b579ae7aa5b3ba7998ca31ddfef58b94)
where
![{\displaystyle [s,t;x,y]_{n}=\sum _{k=0}^{n}(-1)^{k}{s \choose k}{t \choose {n-k}}B_{n-k}(x)B_{k}(y).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8fd533d20dd6ab767f9af30cc2d3cf69a3aacc77)
Note the simple large n limit to suitably scaled trigonometric functions.
![{\displaystyle {\begin{aligned}C_{\nu }(x)&=\sum _{k=0}^{\infty }{\frac {\cos((2k+1)\pi x)}{(2k+1)^{\nu }}}\\[3mu]S_{\nu }(x)&=\sum _{k=0}^{\infty }{\frac {\sin((2k+1)\pi x)}{(2k+1)^{\nu }}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6193e6c07057a79e532ba06550fc7c451c60b382)
for 
, the Euler polynomial has the Fourier series
![{\displaystyle {\begin{aligned}C_{2n}(x)&={\frac {\left(-1\right)^{n}}{4(2n-1)!}}\pi ^{2n}E_{2n-1}(x)\\[1ex]S_{2n+1}(x)&={\frac {\left(-1\right)^{n}}{4(2n)!}}\pi ^{2n+1}E_{2n}(x).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd68fcbc89a2c76a1c7a675da3b19d8cfb066fc9)
Note that the 
and 
are odd and even, respectively:
and
where 
and
denotes the Stirling number of the second kind. The above may be inverted to express the falling factorial in terms of the Bernoulli polynomials:
![{\displaystyle (x)_{n+1}=\sum _{k=0}^{n}{\frac {n+1}{k+1}}\left[{\begin{matrix}n\\k\end{matrix}}\right]\left(B_{k+1}(x)-B_{k+1}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa131f8f6fb91cb75224a8f148cd9a3cb1a7b402)
where
![{\displaystyle \left[{\begin{matrix}n\\k\end{matrix}}\right]=s(n,k)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03a86615f2217e1bcb2401407be4f1261e8da424)
denotes the Stirling number of the first kind.
![{\displaystyle {\begin{aligned}E_{n}(mx)&=m^{n}\sum _{k=0}^{m-1}\left(-1\right)^{k}E_{n}{\left(x+{\frac {k}{m}}\right)}&{\text{ for odd }}m\\[1ex]E_{n}(mx)&={\frac {-2}{n+1}}m^{n}\sum _{k=0}^{m-1}\left(-1\right)^{k}B_{n+1}{\left(x+{\frac {k}{m}}\right)}&{\text{ for even }}m\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3fe434cc17442484737617e19787588c2b587824)
is continuous for all 
exists and is continuous for 
for 
