Polynomial sequence
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 -axis in the
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
The Bernoulli polynomials B n can be defined by a
generating function . They also admit a variety of derived representations.
Generating functions
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!}}.}
The generating function for the Euler polynomials is
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
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}.}
for
n ≥ 0, where
B k are the
Bernoulli numbers , and
E k are the
Euler numbers .
Representation by a differential operator
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}}
where
D =
d /
dx is differentiation with respect to
x and the fraction is expanded as a
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}}.}
cf.
§ 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
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}
on polynomials
f , simply amounts to
(
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}}}
This can be used to produce the
inversion formulae below .
Integral Recurrence
In,
[1]
[2] it is deduced and proved that the Bernoulli polynomials can be obtained by the following integral recurrence
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
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)}
where
ζ
(
s
,
q
)
{\displaystyle \zeta (s,\,q)}
is the
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},}
that is,
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}}
where
Δ
{\displaystyle \Delta }
is the
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}
where
D is differentiation with respect to
x , we have, from the
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 th-degree polynomial such as
x
m
,
{\displaystyle x^{m},}
one may let n go from 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 th powers
Using either the above
integral representation of
x
n
{\displaystyle x^{n}}
or the
identity
B
n
(
x
+
1
)
−
B
n
(
x
)
=
n
x
n
−
1
{\displaystyle B_{n}(x+1)-B_{n}(x)=nx^{n-1}}
, we have
∑
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}}}
(assuming 0
0 = 1).
The Bernoulli and Euler numbers
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}}
for
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,}
since
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
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
At higher n the amount of variation in
B
n
(
x
)
{\displaystyle B_{n}(x)}
between
x
=
0
{\displaystyle x=0}
and
x
=
1
{\displaystyle x=1}
gets large. For instance,
B
16
(
0
)
=
B
16
(
1
)
=
{\displaystyle B_{16}(0)=B_{16}(1)={}}
−
3617
510
≈
−
7.09
,
{\displaystyle -{\tfrac {3617}{510}}\approx -7.09,}
but
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] showed that the maximum value (Mn ) of
B
n
(
x
)
{\displaystyle B_{n}(x)}
between 0 and 1 obeys
M
n
<
2
n
!
(
2
π
)
n
{\displaystyle M_{n}<{\frac {2n!}{(2\pi )^{n}}}}
unless
n is
2 modulo 4 , in which case
M
n
=
2
ζ
(
n
)
n
!
(
2
π
)
n
{\displaystyle M_{n}={\frac {2\zeta (n)\,n!}{(2\pi )^{n}}}}
(where
ζ
(
x
)
{\displaystyle \zeta (x)}
is the
Riemann zeta function ), while the minimum (
mn ) obeys
m
n
>
−
2
n
!
(
2
π
)
n
{\displaystyle m_{n}>{\frac {-2n!}{(2\pi )^{n}}}}
unless
n = 0 modulo 4 , 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
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}.}
These
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
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}}}
These identities are also equivalent to saying that these polynomial sequences are
Appell sequences . (
Hermite polynomials are another example.)
Symmetries
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] established the following surprising symmetry relation: If
r + s + t = n and
x + y + z = 1, then
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,}
where
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
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}}}.}
Note the simple large
n limit to suitably scaled trigonometric functions.
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 ≤ 1 when n ≥ 2 and is valid for 0 < x < 1 when n = 1 .
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}}}
for
ν
>
1
{\displaystyle \nu >1}
, the Euler polynomial has the Fourier series
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}}}
Note that the
C
ν
{\displaystyle C_{\nu }}
and
S
ν
{\displaystyle S_{\nu }}
are odd and even, respectively:
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
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)}
and
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
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}}
where
B
n
=
B
n
(
0
)
{\displaystyle B_{n}=B_{n}(0)}
and
{
n
k
}
=
S
(
n
,
k
)
{\displaystyle \left\{{\begin{matrix}n\\k\end{matrix}}\right\}=S(n,k)}
denotes the
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)}
where
n
k
=
s
(
n
,
k
)
{\displaystyle \left[{\begin{matrix}n\\k\end{matrix}}\right]=s(n,k)}
denotes the
Stirling number of the first kind .
Multiplication theorems
The
multiplication theorems were given by
Joseph Ludwig Raabe in 1851:
For a natural number m ≥1 ,
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
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
A periodic Bernoulli polynomial P n (x ) is a Bernoulli polynomial evaluated at the
fractional part of the argument x . These functions are used to provide the
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 ) is not even a function, being the derivative of a sawtooth and so a
Dirac comb .
The following properties are of interest, valid for all
x
{\displaystyle x}
:
P
k
(
x
)
{\displaystyle P_{k}(x)}
is continuous for all
k
>
1
{\displaystyle k>1}
P
k
′
(
x
)
{\displaystyle P_{k}'(x)}
exists and is continuous for
k
>
2
{\displaystyle k>2}
P
k
′
(
x
)
=
k
P
k
−
1
(
x
)
{\displaystyle P'_{k}(x)=kP_{k-1}(x)}
for
k
>
2
{\displaystyle k>2}
See also
References
Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , (1972) Dover, New York. (See Chapter 23)
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 , Cambridge University Press,
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
International National Other