# Bernoulli polynomials

In mathematics, the Bernoulli polynomials occur in the study of many special functions and in particular the Riemann zeta function and the Hurwitz zeta function. This is in large part because they are an Appell sequence, i.e. a Sheffer sequence for the ordinary derivative operator. Unlike orthogonal polynomials, the Bernoulli polynomials are remarkable in that the number of crossings of the x-axis in the unit interval does not go up as the degree of the polynomials goes up. In the limit of large degree, the Bernoulli polynomials, appropriately scaled, approach the sine and cosine functions.

Bernoulli polynomials

## Representations

The Bernoulli polynomials Bn admit a variety of different representations. Which among them should be taken to be the definition may depend on one's purposes.

### Explicit formula

${\displaystyle B_{n}(x)=\sum _{k=0}^{n}{n \choose k}b_{n-k}x^{k},}$

for n ≥ 0, where bk are the Bernoulli numbers.

### Generating functions

The generating function for the Bernoulli polynomials is

${\displaystyle {\frac {te^{xt}}{e^{t}-1}}=\sum _{n=0}^{\infty }B_{n}(x){\frac {t^{n}}{n!}}.}$

The generating function for the Euler polynomials is

${\displaystyle {\frac {2e^{xt}}{e^{t}+1}}=\sum _{n=0}^{\infty }E_{n}(x){\frac {t^{n}}{n!}}.}$

### Representation by a differential operator

The Bernoulli polynomials are also given by

${\displaystyle B_{n}(x)={D \over e^{D}-1}x^{n}}$

where D = d/dx is differentiation with respect to x and the fraction is expanded as a formal power series. It follows that

${\displaystyle \int _{a}^{x}B_{n}(u)~du={\frac {B_{n+1}(x)-B_{n+1}(a)}{n+1}}~.}$

cf. integrals below.

### Representation by an integral operator

The Bernoulli polynomials are the unique polynomials determined by

${\displaystyle \int _{x}^{x+1}B_{n}(u)\,du=x^{n}.}$
${\displaystyle (Tf)(x)=\int _{x}^{x+1}f(u)\,du}$

on polynomials f, simply amounts to

{\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

${\displaystyle B_{m}(x)=\sum _{n=0}^{m}{\frac {1}{n+1}}\sum _{k=0}^{n}(-1)^{k}{n \choose k}(x+k)^{m}.}$

Note the remarkable similarity to the globally convergent series expression for the Hurwitz zeta function. Indeed, one has

${\displaystyle B_{n}(x)=-n\zeta (1-n,x)}$

where ζ(sq) is the Hurwitz zeta; thus, in a certain sense, the Hurwitz zeta generalizes the Bernoulli polynomials to non-integer values of n.

The inner sum may be understood to be the nth forward difference of xm; that is,

${\displaystyle \Delta ^{n}x^{m}=\sum _{k=0}^{n}(-1)^{n-k}{n \choose k}(x+k)^{m}}$

where Δ is the forward difference operator. Thus, one may write

${\displaystyle B_{m}(x)=\sum _{n=0}^{m}{\frac {(-1)^{n}}{n+1}}\Delta ^{n}x^{m}.}$

This formula may be derived from an identity appearing above as follows. Since the forward difference operator Δ equals

${\displaystyle \Delta =e^{D}-1\,}$

where D is differentiation with respect to x, we have, from the Mercator series

${\displaystyle {D \over e^{D}-1}={\log(\Delta +1) \over \Delta }=\sum _{n=0}^{\infty }{(-\Delta )^{n} \over n+1}.}$

As long as this operates on an mth-degree polynomial such as xm, 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

${\displaystyle E_{m}(x)=\sum _{n=0}^{m}{\frac {1}{2^{n}}}\sum _{k=0}^{n}(-1)^{k}{n \choose k}(x+k)^{m}\,.}$

This may also be written in terms of the Euler numbers Ek as

${\displaystyle E_{m}(x)=\sum _{k=0}^{m}{m \choose k}{\frac {E_{k}}{2^{k}}}\left(x-{\frac {1}{2}}\right)^{m-k}\,.}$

## Sums of pth powers

We have

${\displaystyle \sum _{k=0}^{x}k^{p}={\frac {B_{p+1}(x+1)-B_{p+1}(0)}{p+1}}}$

(assuming 00=1). See Faulhaber's formula for more on this.

## The Bernoulli and Euler numbers

The Bernoulli numbers are given by ${\displaystyle B_{n}=B_{n}(0).}$ An alternate convention defines the Bernoulli numbers as ${\displaystyle B_{n}=B_{n}(1)}$. This definition gives Bn = −nζ(1 − n) where for n = 0 and n = 1 the expression −nζ(1 − n) is to be understood as limx → n −xζ(1 − x). The two conventions differ only for n = 1 since B1(1) = 1/2 = −B1(0).

## Explicit expressions for low degrees

The first few Bernoulli polynomials are:

${\displaystyle B_{0}(x)=1\,}$
${\displaystyle B_{1}(x)=x-1/2\,}$
${\displaystyle B_{2}(x)=x^{2}-x+1/6\,}$
${\displaystyle B_{3}(x)=x^{3}-{\frac {3}{2}}x^{2}+{\frac {1}{2}}x\,}$
${\displaystyle B_{4}(x)=x^{4}-2x^{3}+x^{2}-{\frac {1}{30}}\,}$
${\displaystyle B_{5}(x)=x^{5}-{\frac {5}{2}}x^{4}+{\frac {5}{3}}x^{3}-{\frac {1}{6}}x\,}$
${\displaystyle B_{6}(x)=x^{6}-3x^{5}+{\frac {5}{2}}x^{4}-{\frac {1}{2}}x^{2}+{\frac {1}{42}}.\,}$

The first few Euler polynomials are

${\displaystyle E_{0}(x)=1\,}$
${\displaystyle E_{1}(x)=x-1/2\,}$
${\displaystyle E_{2}(x)=x^{2}-x\,}$
${\displaystyle E_{3}(x)=x^{3}-{\frac {3}{2}}x^{2}+{\frac {1}{4}}\,}$
${\displaystyle E_{4}(x)=x^{4}-2x^{3}+x\,}$
${\displaystyle E_{5}(x)=x^{5}-{\frac {5}{2}}x^{4}+{\frac {5}{2}}x^{2}-{\frac {1}{2}}\,}$
${\displaystyle E_{6}(x)=x^{6}-3x^{5}+5x^{3}-3x.\,}$

## Maximum and minimum

At higher n, the amount of variation in Bn(x) between x = 0 and x = 1 gets large. For instance,

${\displaystyle B_{16}(x)=x^{16}-8x^{15}+20x^{14}-{\frac {182}{3}}x^{12}+{\frac {572}{3}}x^{10}-429x^{8}+{\frac {1820}{3}}x^{6}-{\frac {1382}{3}}x^{4}+140x^{2}-{\frac {3617}{510}}}$

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 Bn(x) between 0 and 1 obeys

${\displaystyle M_{n}<{\frac {2n!}{(2\pi )^{n}}}}$

unless n is 2 modulo 4, in which case

${\displaystyle M_{n}={\frac {2\zeta (n)n!}{(2\pi )^{n}}}}$

(where ${\displaystyle \zeta (x)}$ is the Riemann zeta function), while the minimum obeys

${\displaystyle m_{n}>{\frac {-2n!}{(2\pi )^{n}}}}$

unless n is 0 modulo 4, in which case

${\displaystyle m_{n}={\frac {-2\zeta (n)n!}{(2\pi )^{n}}}.}$

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:

${\displaystyle \Delta B_{n}(x)=B_{n}(x+1)-B_{n}(x)=nx^{n-1},\,}$
${\displaystyle \Delta E_{n}(x)=E_{n}(x+1)-E_{n}(x)=2(x^{n}-E_{n}(x)).\,}$

(Δ is the forward difference operator).

These polynomial sequences are Appell sequences:

${\displaystyle B_{n}'(x)=nB_{n-1}(x),\,}$
${\displaystyle E_{n}'(x)=nE_{n-1}(x).\,}$

### Translations

${\displaystyle B_{n}(x+y)=\sum _{k=0}^{n}{n \choose k}B_{k}(x)y^{n-k}}$
${\displaystyle E_{n}(x+y)=\sum _{k=0}^{n}{n \choose k}E_{k}(x)y^{n-k}}$

These identities are also equivalent to saying that these polynomial sequences are Appell sequences. (Hermite polynomials are another example.)

### Symmetries

${\displaystyle B_{n}(1-x)=(-1)^{n}B_{n}(x),\quad n\geq 0,}$
${\displaystyle E_{n}(1-x)=(-1)^{n}E_{n}(x)\,}$
${\displaystyle (-1)^{n}B_{n}(-x)=B_{n}(x)+nx^{n-1}\,}$
${\displaystyle (-1)^{n}E_{n}(-x)=-E_{n}(x)+2x^{n}\,}$

Zhi-Wei Sun and Hao Pan [2] established the following surprising symmetry relation: If r + s + t = n and x + y + z = 1, then

${\displaystyle r[s,t;x,y]_{n}+s[t,r;y,z]_{n}+t[r,s;z,x]_{n}=0,}$

where

${\displaystyle [s,t;x,y]_{n}=\sum _{k=0}^{n}(-1)^{k}{s \choose k}{t \choose {n-k}}B_{n-k}(x)B_{k}(y).}$

## Fourier series

The Fourier series of the Bernoulli polynomials is also a Dirichlet series, given by the expansion

${\displaystyle B_{n}(x)=-{\frac {n!}{(2\pi i)^{n}}}\sum _{k\not =0}{\frac {e^{2\pi ikx}}{k^{n}}}=-2n!\sum _{k=1}^{\infty }{\frac {\cos \left(2k\pi x-{\frac {n\pi }{2}}\right)}{(2k\pi )^{n}}}.}$

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

${\displaystyle B_{n}(x)=-\Gamma (n+1)\sum _{k=1}^{\infty }{\frac {\exp(2\pi ikx)+e^{i\pi n}\exp(2\pi ik(1-x))}{(2\pi ik)^{n}}}.}$

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

${\displaystyle C_{\nu }(x)=\sum _{k=0}^{\infty }{\frac {\cos((2k+1)\pi x)}{(2k+1)^{\nu }}}}$

and

${\displaystyle S_{\nu }(x)=\sum _{k=0}^{\infty }{\frac {\sin((2k+1)\pi x)}{(2k+1)^{\nu }}}}$

for ${\displaystyle \nu >1}$, the Euler polynomial has the Fourier series

${\displaystyle C_{2n}(x)={\frac {(-1)^{n}}{4(2n-1)!}}\pi ^{2n}E_{2n-1}(x)}$

and

${\displaystyle S_{2n+1}(x)={\frac {(-1)^{n}}{4(2n)!}}\pi ^{2n+1}E_{2n}(x).}$

Note that the ${\displaystyle C_{\nu }}$ and ${\displaystyle S_{\nu }}$ are odd and even, respectively:

${\displaystyle C_{\nu }(x)=-C_{\nu }(1-x)}$

and

${\displaystyle S_{\nu }(x)=S_{\nu }(1-x).}$

They are related to the Legendre chi function ${\displaystyle \chi _{\nu }}$ as

${\displaystyle C_{\nu }(x)={\mbox{Re}}\chi _{\nu }(e^{ix})}$

and

${\displaystyle S_{\nu }(x)={\mbox{Im}}\chi _{\nu }(e^{ix}).}$

## 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 #Representation by an integral operator, it follows that

${\displaystyle x^{n}={\frac {1}{n+1}}\sum _{k=0}^{n}{n+1 \choose k}B_{k}(x)}$

and

${\displaystyle x^{n}=E_{n}(x)+{\frac {1}{2}}\sum _{k=0}^{n-1}{n \choose k}E_{k}(x).}$

## Relation to falling factorial

The Bernoulli polynomials may be expanded in terms of the falling factorial ${\displaystyle (x)_{k}}$ as

${\displaystyle B_{n+1}(x)=B_{n+1}+\sum _{k=0}^{n}{\frac {n+1}{k+1}}\left\{{\begin{matrix}n\\k\end{matrix}}\right\}(x)_{k+1}}$
${\displaystyle \left\{{\begin{matrix}n\\k\end{matrix}}\right\}=S(n,k)}$

denotes the Stirling number of the second kind. The above may be inverted to express the falling factorial in terms of the Bernoulli polynomials:

${\displaystyle (x)_{n+1}=\sum _{k=0}^{n}{\frac {n+1}{k+1}}\left[{\begin{matrix}n\\k\end{matrix}}\right]\left(B_{k+1}(x)-B_{k+1}\right)}$

where

${\displaystyle \left[{\begin{matrix}n\\k\end{matrix}}\right]=s(n,k)}$

denotes the Stirling number of the first kind.

## Multiplication theorems

The multiplication theorems were given by Joseph Ludwig Raabe in 1851:

${\displaystyle B_{n}(mx)=m^{n-1}\sum _{k=0}^{m-1}B_{n}\left(x+{\frac {k}{m}}\right)}$
${\displaystyle E_{n}(mx)=m^{n}\sum _{k=0}^{m-1}(-1)^{k}E_{n}\left(x+{\frac {k}{m}}\right)\quad {\mbox{ for }}m=1,3,\dots }$
${\displaystyle E_{n}(mx)={\frac {-2}{n+1}}m^{n}\sum _{k=0}^{m-1}(-1)^{k}B_{n+1}\left(x+{\frac {k}{m}}\right)\quad {\mbox{ for }}m=2,4,\dots }$

## Integrals

Indefinite integrals

${\displaystyle \int _{a}^{x}B_{n}(t)\,dt={\frac {B_{n+1}(x)-B_{n+1}(a)}{n+1}}}$
${\displaystyle \int _{a}^{x}E_{n}(t)\,dt={\frac {E_{n+1}(x)-E_{n+1}(a)}{n+1}}}$

Definite integrals

${\displaystyle \int _{0}^{1}B_{n}(t)B_{m}(t)\,dt=(-1)^{n-1}{\frac {m!n!}{(m+n)!}}B_{n+m}\quad {\mbox{ for }}m,n\geq 1}$
${\displaystyle \int _{0}^{1}E_{n}(t)E_{m}(t)\,dt=(-1)^{n}4(2^{m+n+2}-1){\frac {m!n!}{(m+n+2)!}}B_{n+m+2}}$

## Periodic Bernoulli polynomials

A periodic Bernoulli polynomial Pn(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.

## References

1. D.H. Lehmer, "On the Maxima and Minima of Bernoulli Polynomials", American Mathematical Monthly, volume 47, pages 533–538 (1940)
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