Polynomial sequence
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-{\frac {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)={D \over 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 .
Another explicit formula [ ]
An explicit formula for the Bernoulli polynomials is given by
B
m
(
x
)
=
∑
n
=
0
m
1
n
+
1
∑
k
=
0
n
(
−
1
)
k
(
n
k
)
(
x
+
k
)
m
.
{\displaystyle B_{m}(x)=\sum _{n=0}^{m}{\frac {1}{n+1}}\sum _{k=0}^{n}(-1)^{k}{n \choose k}(x+k)^{m}.}
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 ) 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 ; 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 is the forward difference operator . Thus, one may write
B
m
(
x
)
=
∑
n
=
0
m
(
−
1
)
n
n
+
1
Δ
n
x
m
.
{\displaystyle B_{m}(x)=\sum _{n=0}^{m}{\frac {(-1)^{n}}{n+1}}\,\Delta ^{n}x^{m}.}
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 {D \over e^{D}-1}={\log(\Delta +1) \over \Delta }=\sum _{n=0}^{\infty }{(-\Delta )^{n} \over n+1}.}
As long as this operates on an m th-degree polynomial such as 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
m
(
x
)
=
∑
n
=
0
m
1
2
n
∑
k
=
0
n
(
−
1
)
k
(
n
k
)
(
x
+
k
)
m
.
{\displaystyle E_{m}(x)=\sum _{n=0}^{m}{\frac {1}{2^{n}}}\sum _{k=0}^{n}(-1)^{k}{n \choose k}(x+k)^{m}\,.}
The above follows analogously, using the fact that
2
e
D
+
1
=
1
1
+
Δ
/
2
=
∑
n
=
0
∞
(
−
Δ
2
)
n
.
{\displaystyle {\frac {2}{e^{D}+1}}={\frac {1}{1+\Delta /2}}=\sum _{n=0}^{\infty }{\Bigl (}-{\frac {\Delta }{2}}{\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 00 = 1).
The Bernoulli and Euler numbers [ ]
The Bernoulli numbers are given by
B
n
=
B
n
(
0
)
.
{\displaystyle \textstyle B_{n}=B_{n}(0).}
This definition gives
ζ
(
−
n
)
=
(
−
1
)
n
n
+
1
B
n
+
1
{\displaystyle \textstyle \zeta (-n)={\frac {(-1)^{n}}{n+1}}B_{n+1}}
for
n
=
0
,
1
,
2
,
…
{\displaystyle \textstyle n=0,1,2,\ldots }
.
An alternate convention defines the Bernoulli numbers as
B
n
=
B
n
(
1
)
.
{\displaystyle \textstyle B_{n}=B_{n}(1).}
The two conventions differ only for
n
=
1
{\displaystyle n=1}
since
B
1
(
1
)
=
1
2
=
−
B
1
(
0
)
{\displaystyle B_{1}(1)={\tfrac {1}{2}}=-B_{1}(0)}
.
The Euler numbers are given by
E
n
=
2
n
E
n
(
1
2
)
.
{\displaystyle E_{n}=2^{n}E_{n}({\tfrac {1}{2}}).}
Explicit expressions for low degrees [ ]
The first few Bernoulli polynomials are:
B
0
(
x
)
=
1
B
1
(
x
)
=
x
−
1
2
B
2
(
x
)
=
x
2
−
x
+
1
6
B
3
(
x
)
=
x
3
−
3
2
x
2
+
1
2
x
B
4
(
x
)
=
x
4
−
2
x
3
+
x
2
−
1
30
B
5
(
x
)
=
x
5
−
5
2
x
4
+
5
3
x
3
−
1
6
x
B
6
(
x
)
=
x
6
−
3
x
5
+
5
2
x
4
−
1
2
x
2
+
1
42
.
{\displaystyle {\begin{aligned}B_{0}(x)&=1\\[8pt]B_{1}(x)&=x-{\frac {1}{2}}\\[8pt]B_{2}(x)&=x^{2}-x+{\frac {1}{6}}\\[8pt]B_{3}(x)&=x^{3}-{\frac {3}{2}}x^{2}+{\frac {1}{2}}x\\[8pt]B_{4}(x)&=x^{4}-2x^{3}+x^{2}-{\frac {1}{30}}\\[8pt]B_{5}(x)&=x^{5}-{\frac {5}{2}}x^{4}+{\frac {5}{3}}x^{3}-{\frac {1}{6}}x\\[8pt]B_{6}(x)&=x^{6}-3x^{5}+{\frac {5}{2}}x^{4}-{\frac {1}{2}}x^{2}+{\frac {1}{42}}.\end{aligned}}}
The first few Euler polynomials are:
E
0
(
x
)
=
1
E
1
(
x
)
=
x
−
1
2
E
2
(
x
)
=
x
2
−
x
E
3
(
x
)
=
x
3
−
3
2
x
2
+
1
4
E
4
(
x
)
=
x
4
−
2
x
3
+
x
E
5
(
x
)
=
x
5
−
5
2
x
4
+
5
2
x
2
−
1
2
E
6
(
x
)
=
x
6
−
3
x
5
+
5
x
3
−
3
x
.
{\displaystyle {\begin{aligned}E_{0}(x)&=1\\[8pt]E_{1}(x)&=x-{\frac {1}{2}}\\[8pt]E_{2}(x)&=x^{2}-x\\[8pt]E_{3}(x)&=x^{3}-{\frac {3}{2}}x^{2}+{\frac {1}{4}}\\[8pt]E_{4}(x)&=x^{4}-2x^{3}+x\\[8pt]E_{5}(x)&=x^{5}-{\frac {5}{2}}x^{4}+{\frac {5}{2}}x^{2}-{\frac {1}{2}}\\[8pt]E_{6}(x)&=x^{6}-3x^{5}+5x^{3}-3x.\end{aligned}}}
Maximum and minimum [ ]
At higher n , the amount of variation in B n (x ) between x = 0 and x = 1 gets large. For instance,
B
16
(
x
)
=
x
16
−
8
x
15
+
20
x
14
−
182
3
x
12
+
572
3
x
10
−
429
x
8
+
1820
3
x
6
−
1382
3
x
4
+
140
x
2
−
3617
510
{\displaystyle B_{16}(x)=x^{16}-8x^{15}+20x^{14}-{\frac {182}{3}}x^{12}+{\frac {572}{3}}x^{10}-429x^{8}+{\frac {1820}{3}}x^{6}-{\frac {1382}{3}}x^{4}+140x^{2}-{\frac {3617}{510}}}
which shows that the value at x = 0 (and at x = 1) is −3617/510 ≈ −7.09, while at x = 1/2, the value is 118518239/3342336 ≈ +7.09. D.H. Lehmer [1] showed that the maximum value of 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 obeys
m
n
>
−
2
n
!
(
2
π
)
n
{\displaystyle m_{n}>{\frac {-2n!}{(2\pi )^{n}}}}
unless n is 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
,
{\displaystyle \Delta B_{n}(x)=B_{n}(x+1)-B_{n}(x)=nx^{n-1},}
Δ
E
n
(
x
)
=
E
n
(
x
+
1
)
−
E
n
(
x
)
=
2
(
x
n
−
E
n
(
x
)
)
.
{\displaystyle \Delta E_{n}(x)=E_{n}(x+1)-E_{n}(x)=2(x^{n}-E_{n}(x)).}
(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
)
,
{\displaystyle B_{n}'(x)=nB_{n-1}(x),}
E
n
′
(
x
)
=
n
E
n
−
1
(
x
)
.
{\displaystyle E_{n}'(x)=nE_{n-1}(x).}
Translations [ ]
B
n
(
x
+
y
)
=
∑
k
=
0
n
(
n
k
)
B
k
(
x
)
y
n
−
k
{\displaystyle B_{n}(x+y)=\sum _{k=0}^{n}{n \choose k}B_{k}(x)y^{n-k}}
E
n
(
x
+
y
)
=
∑
k
=
0
n
(
n
k
)
E
k
(
x
)
y
n
−
k
{\displaystyle E_{n}(x+y)=\sum _{k=0}^{n}{n \choose k}E_{k}(x)y^{n-k}}
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
,
{\displaystyle B_{n}(1-x)=(-1)^{n}B_{n}(x),\quad n\geq 0,}
E
n
(
1
−
x
)
=
(
−
1
)
n
E
n
(
x
)
{\displaystyle E_{n}(1-x)=(-1)^{n}E_{n}(x)}
(
−
1
)
n
B
n
(
−
x
)
=
B
n
(
x
)
+
n
x
n
−
1
{\displaystyle (-1)^{n}B_{n}(-x)=B_{n}(x)+nx^{n-1}}
(
−
1
)
n
E
n
(
−
x
)
=
−
E
n
(
x
)
+
2
x
n
{\displaystyle (-1)^{n}E_{n}(-x)=-E_{n}(x)+2x^{n}}
B
n
(
1
2
)
=
(
1
2
n
−
1
−
1
)
B
n
,
n
≥
0
from the multiplication theorems below.
{\displaystyle B_{n}\left({\frac {1}{2}}\right)=\left({\frac {1}{2^{n-1}}}-1\right)B_{n},\quad n\geq 0{\text{ from the multiplication theorems below.}}}
Zhi-Wei Sun and Hao Pan [2] 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
{\displaystyle C_{\nu }(x)=\sum _{k=0}^{\infty }{\frac {\cos((2k+1)\pi x)}{(2k+1)^{\nu }}}}
and
S
ν
(
x
)
=
∑
k
=
0
∞
sin
(
(
2
k
+
1
)
π
x
)
(
2
k
+
1
)
ν
{\displaystyle S_{\nu }(x)=\sum _{k=0}^{\infty }{\frac {\sin((2k+1)\pi x)}{(2k+1)^{\nu }}}}
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
)
{\displaystyle C_{2n}(x)={\frac {(-1)^{n}}{4(2n-1)!}}\pi ^{2n}E_{2n-1}(x)}
and
S
2
n
+
1
(
x
)
=
(
−
1
)
n
4
(
2
n
)
!
π
2
n
+
1
E
2
n
(
x
)
.
{\displaystyle S_{2n+1}(x)={\frac {(-1)^{n}}{4(2n)!}}\pi ^{2n+1}E_{2n}(x).}
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
)
{\displaystyle C_{\nu }(x)=-C_{\nu }(1-x)}
and
S
ν
(
x
)
=
S
ν
(
1
−
x
)
.
{\displaystyle S_{\nu }(x)=S_{\nu }(1-x).}
They are related to the Legendre chi function
χ
ν
{\displaystyle \chi _{\nu }}
as
C
ν
(
x
)
=
Re
χ
ν
(
e
i
x
)
{\displaystyle C_{\nu }(x)=\operatorname {Re} \chi _{\nu }(e^{ix})}
and
S
ν
(
x
)
=
Im
χ
ν
(
e
i
x
)
.
{\displaystyle S_{\nu }(x)=\operatorname {Im} \chi _{\nu }(e^{ix}).}
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
m
=
1
,
3
,
…
{\displaystyle E_{n}(mx)=m^{n}\sum _{k=0}^{m-1}(-1)^{k}E_{n}\left(x+{\frac {k}{m}}\right)\quad {\mbox{ for }}m=1,3,\dots }
E
n
(
m
x
)
=
−
2
n
+
1
m
n
∑
k
=
0
m
−
1
(
−
1
)
k
B
n
+
1
(
x
+
k
m
)
for
m
=
2
,
4
,
…
{\displaystyle E_{n}(mx)={\frac {-2}{n+1}}m^{n}\sum _{k=0}^{m-1}(-1)^{k}B_{n+1}\left(x+{\frac {k}{m}}\right)\quad {\mbox{ for }}m=2,4,\dots }
Integrals [ ]
Two definite integrals relating the Bernoulli and Euler polynomials to the Bernoulli and Euler numbers are:[citation needed ]
∫
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}}
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
)
is continuous for all
k
>
1
P
k
′
(
x
)
exists and is continuous for
k
>
2
P
k
′
(
x
)
=
k
P
k
−
1
(
x
)
,
k
>
2
{\displaystyle {\begin{aligned}&P_{k}(x){\text{ is continuous for all }}k>1\\[5pt]&P_{k}'(x){\text{ exists and is continuous for }}k>2\\[5pt]&P'_{k}(x)=kP_{k-1}(x),k>2\end{aligned}}}
See also [ ]
References [ ]
^ D.H. Lehmer, "On the Maxima and Minima of Bernoulli Polynomials", American Mathematical Monthly , volume 47, pages 533–538 (1940)
^ Zhi-Wei Sun; Hao Pan (2006). "Identities concerning Bernoulli and Euler polynomials". Acta Arithmetica . 125 : 21–39. arXiv :math/0409035 . Bibcode :2006AcAri.125...21S . doi :10.4064/aa125-1-3 .
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 : 1527–1535. doi :10.2307/2161144 .
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 . (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. 97 . Cambridge: Cambridge Univ. Press. pp. 495–519. ISBN 0-521-84903-9 .
External links [ ]
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