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 .
The Bernoulli polynomials B n can be defined by a 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!}}.} 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!}}.}
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 [ 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}} 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 [ 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} 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 [ edit ] 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.}
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}.}
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).
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 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 [ 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}.} 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}}}
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.)
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).}
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}}}
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 [ 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}} 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 [ edit ] 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}}}
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 ) 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} 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 .
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