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
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]
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},}
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},} 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}} 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}}} (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}} 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 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)} 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] Mn ) of B n ( x ) {\displaystyle B_{n}(x)} 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 ≥ 20 < 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 }}
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}}
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 )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 References ^ Hurtado Benavides, Miguel Ángel. (2020). De las sumas de potencias a las sucesiones de Appell y su caracterización a través de funcionales. [Tesis de maestría]. Universidad Sergio Arboleda. https://repository.usergioarboleda.edu.co/handle/11232/174 ^ Sergio A. Carrillo; Miguel Hurtado. Appell and Sheffer sequences: on their characterizations through functionals and examples. Comptes Rendus. Mathématique, Tome 359 (2021) no. 2, pp. 205-217. doi : 10.5802/crmath.172. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.172/ ^ Lehmer, D.H. (1940). "On the maxima and minima of Bernoulli polynomials". American Mathematical Monthly 47 : 533–538.^ 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. Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables(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 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 A list of integral identities involving Bernoulli polynomials from NIST
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![{\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)
![{\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)
![{\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)
![{\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)
![{\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)
![{\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)
![{\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)
![{\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)
![{\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)
![{\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)
![{\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)
![{\displaystyle \left[{\begin{matrix}n\\k\end{matrix}}\right]=s(n,k)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03a86615f2217e1bcb2401407be4f1261e8da424)
![{\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)
