# Bernoulli polynomials

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Polynomial sequence

Bernoulli polynomials

In [mathematics](/source/Mathematics), the **Bernoulli polynomials**, named after [Jacob Bernoulli](/source/Jacob_Bernoulli), combine the [Bernoulli numbers](/source/Bernoulli_number) and [binomial coefficients](/source/Binomial_coefficient). They are used for [series expansion](/source/Series_expansion) of [functions](/source/Function_(mathematics)), and with the [Euler–MacLaurin formula](/source/Euler%E2%80%93MacLaurin_formula).

These [polynomials](/source/Polynomial) occur in the study of many [special functions](/source/Special_functions) and, in particular, the [Riemann zeta function](/source/Riemann_zeta_function) and the [Hurwitz zeta function](/source/Hurwitz_zeta_function). They are an [Appell sequence](/source/Appell_sequence) (i.e. a [Sheffer sequence](/source/Sheffer_sequence) for the ordinary [derivative](/source/Derivative) operator). For the Bernoulli polynomials, the number of crossings of the *x*-axis in the [unit interval](/source/Unit_interval) does not go up with the [degree](/source/Degree_of_a_polynomial). In the limit of large degree, they approach, when appropriately scaled, the [sine and cosine functions](/source/Trigonometric_function).

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](/source/Generating_function). They also admit a variety of derived representations.

### Generating functions

The [exponential generating function](/source/Exponential_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 exponential 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 {\displaystyle n\geq 0} , where B k {\displaystyle B_{k}} are the [Bernoulli numbers](/source/Bernoulli_number), and E k {\displaystyle E_{k}} are the [Euler numbers](/source/Euler_numbers). It follows that B n ( 0 ) = B n {\displaystyle B_{n}(0)=B_{n}} and E m ( 1 2 ) = 1 2 m E m {\displaystyle E_{m}{\big (}{\tfrac {1}{2}}{\big )}={\tfrac {1}{2^{m}}}E_{m}} .

### 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 d x {\displaystyle \ D\equiv {\frac {\mathrm {d} }{\ \mathrm {d} x\ }}\ } is differentiation with respect to x and the fraction is expanded as a [formal power series](/source/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)\ \mathrm {d} \ u={\frac {\ B_{n+1}(x)-B_{n+1}(a)\ }{n+1}}~.} cf. [§ Integrals](#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](/source/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](#Inversion).

### 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](/source/Hurwitz_zeta_function) in the [complex plane](/source/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](/source/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 nth [forward difference](/source/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](/source/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](/source/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 mth-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](/source/N%C3%B6rlund%E2%80%93Rice_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

Main article: [Faulhaber's formula](/source/Faulhaber's_formula)

Using either the above [integral representation](#Representation_by_an_integral_operator) of x n {\displaystyle x^{n}} or the [identity](#Differences_and_derivatives) 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

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](/source/D.H._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](/source/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](/source/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](/source/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](/source/Polynomial_sequence) are [Appell sequences](/source/Appell_sequence): 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](/source/Appell_sequence). ([Hermite polynomials](/source/Hermite_polynomials) are another example.)

### Symmetries

B n ( 1 − x ) = ( − 1 ) n B n ( x ) , n ≥ 0 , and in particular for n ≠ 1 , B n ( 0 ) = B n ( 1 ) 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,{\text{ and in particular for }}n\neq 1,~B_{n}(0)=B_{n}(1)\\[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](/source/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](/source/Fourier_series) of the Bernoulli polynomials is also a [Dirichlet series](/source/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](/source/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](/source/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](/source/Monomial) in terms of the polynomials.

Specifically, evidently from the above section on [integral operators](#Representation_by_an_integral_operator), 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](/source/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](/source/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](/source/Stirling_number_of_the_first_kind).

## Multiplication theorems

The [multiplication theorems](/source/Multiplication_theorem) were given by [Joseph Ludwig Raabe](/source/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](/source/Fractional_part) of the argument *x*. These functions are used to provide the [remainder term](/source/Remainder_term) in the [Euler–Maclaurin formula](/source/Euler%E2%80%93Maclaurin_formula) relating sums to integrals. The first polynomial is a [sawtooth function](/source/Sawtooth_wave).

Strictly these functions are not polynomials at all and more properly should be termed the periodic Bernoulli functions.

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

- [Bernoulli numbers](/source/Bernoulli_numbers)

- [Bernoulli polynomials of the second kind](/source/Bernoulli_polynomials_of_the_second_kind)

- [Stirling polynomial](/source/Stirling_polynomial)

- [Polynomials calculating sums of powers of arithmetic progressions](/source/Polynomials_calculating_sums_of_powers_of_arithmetic_progressions)

## References

1. **[^](#cite_ref-1)** 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](https://repository.usergioarboleda.edu.co/handle/11232/174)

1. **[^](#cite_ref-2)** 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/](https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.172/)

1. **[^](#cite_ref-3)** [Lehmer, D.H.](/source/D.H._Lehmer) (1940). "On the maxima and minima of Bernoulli polynomials". *[American Mathematical Monthly](/source/American_Mathematical_Monthly)*. **47** (8): 533–538. [doi](/source/Doi_(identifier)):[10.1080/00029890.1940.11991015](https://doi.org/10.1080%2F00029890.1940.11991015).

1. **[^](#cite_ref-4)** Zhi-Wei Sun; Hao Pan (2006). "Identities concerning Bernoulli and Euler polynomials". *Acta Arithmetica*. **125** (1): 21–39. [arXiv](/source/ArXiv_(identifier)):[math/0409035](https://arxiv.org/abs/math/0409035). [Bibcode](/source/Bibcode_(identifier)):[2006AcAri.125...21S](https://ui.adsabs.harvard.edu/abs/2006AcAri.125...21S). [doi](/source/Doi_(identifier)):[10.4064/aa125-1-3](https://doi.org/10.4064%2Faa125-1-3). [S2CID](/source/S2CID_(identifier)) [10841415](https://api.semanticscholar.org/CorpusID:10841415).

1. **[^](#cite_ref-5)** Takashi Agoh & Karl Dilcher (2011). ["Integrals of products of Bernoulli polynomials"](https://doi.org/10.1016%2Fj.jmaa.2011.03.061). *Journal of Mathematical Analysis and Applications*. **381**: 10–16. [doi](/source/Doi_(identifier)):[10.1016/j.jmaa.2011.03.061](https://doi.org/10.1016%2Fj.jmaa.2011.03.061).

1. **[^](#cite_ref-6)** 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](/source/ArXiv_(identifier)):[1611.01274](https://arxiv.org/abs/1611.01274). [doi](/source/Doi_(identifier)):[10.1080/10652469.2017.1312366](https://doi.org/10.1080%2F10652469.2017.1312366). [S2CID](/source/S2CID_(identifier)) [119132354](https://api.semanticscholar.org/CorpusID:119132354).

- Milton Abramowitz and Irene A. Stegun, eds. *[Handbook of Mathematical Functions](/source/Abramowitz_and_Stegun) with Formulas, Graphs, and Mathematical Tables*, (1972) Dover, New York. *(See Chapter 23)*

- [Apostol, Tom M.](/source/Tom_M._Apostol) (1976), *Introduction to analytic number theory*, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, [ISBN](/source/ISBN_(identifier)) [978-0-387-90163-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-90163-3), [MR](/source/MR_(identifier)) [0434929](https://mathscinet.ams.org/mathscinet-getitem?mr=0434929), [Zbl](/source/Zbl_(identifier)) [0335.10001](https://zbmath.org/?format=complete&q=an:0335.10001) *(See chapter 12.11)*

- Dilcher, K. (2010), ["Bernoulli and Euler Polynomials"](https://dlmf.nist.gov/24), in [Olver, Frank W. J.](/source/Frank_W._J._Olver); Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), *[NIST Handbook of Mathematical Functions](/source/Digital_Library_of_Mathematical_Functions)*, Cambridge University Press, [ISBN](/source/ISBN_(identifier)) [978-0-521-19225-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-19225-5), [MR](/source/MR_(identifier)) [2723248](https://mathscinet.ams.org/mathscinet-getitem?mr=2723248).

- Cvijović, Djurdje; Klinowski, Jacek (1995). ["New formulae for the Bernoulli and Euler polynomials at rational arguments"](https://doi.org/10.1090%2FS0002-9939-1995-1283544-0). *[Proceedings of the American Mathematical Society](/source/Proceedings_of_the_American_Mathematical_Society)*. **123** (5): 1527–1535. [doi](/source/Doi_(identifier)):[10.1090/S0002-9939-1995-1283544-0](https://doi.org/10.1090%2FS0002-9939-1995-1283544-0). [JSTOR](/source/JSTOR_(identifier)) [2161144](https://www.jstor.org/stable/2161144).

- Kouba, Omran (2016). "Lecture Notes, Bernoulli Polynomials and Applications". [arXiv](/source/ArXiv_(identifier)):[1309.7560v2](https://arxiv.org/abs/1309.7560v2) [[math.CA](https://arxiv.org/archive/math.CA)].

- [Jesús Guillera](/source/Jes%C3%BAs_Guillera); 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](/source/ArXiv_(identifier)):[math.NT/0506319](https://arxiv.org/abs/math.NT/0506319). [doi](/source/Doi_(identifier)):[10.1007/s11139-007-9102-0](https://doi.org/10.1007%2Fs11139-007-9102-0). [S2CID](/source/S2CID_(identifier)) [14910435](https://api.semanticscholar.org/CorpusID:14910435). *(Reviews relationship to the Hurwitz zeta function and Lerch transcendent.)*

- [Hugh L. Montgomery](/source/Hugh_Montgomery_(mathematician)); [Robert C. Vaughan](/source/Robert_Charles_Vaughan_(mathematician)) (2007). *Multiplicative number theory I. Classical theory*. Cambridge tracts in advanced mathematics. Vol. 97. Cambridge: Cambridge Univ. Press. pp. 495–519. [ISBN](/source/ISBN_(identifier)) [978-0-521-84903-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-84903-6).

## External links

- [A list of integral identities involving Bernoulli polynomials](https://dlmf.nist.gov/24.7) from [NIST](/source/NIST)

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