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

This article also discusses the Bernoulli polynomials and the related Euler polynomials, and the Bernoulli and Euler numbers.

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

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

\frac{t e^{xt}}{e^t-1}= \sum_{n=0}^\infty B_n(x) \frac{t^n}{n!}.

The generating function for the Euler polynomials is

\frac{2 e^{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

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

\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

\int_x^{x+1} B_n(u)\,du = x^n.

The integral transform

(Tf)(x) = \int_x^{x+1} f(u)\,du

on polynomials f, simply amounts to

\begin{align} (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{align}

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)= \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

B_n(x) = -n \zeta(1-n,x)

where ζ(s, q) 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,

\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)= \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

\Delta = e^D - 1\,

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

{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

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

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

\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 \textstyle B_n=B_n(0).

This definition gives \textstyle \zeta(-n) = -\frac{1}{n+1}B_{n+1} for \textstyle n=0, 1, 2, \ldots.

An alternate convention defines the Bernoulli numbers as \textstyle B_n=B_n(1).

The two conventions differ only for n=1 since B_1(1)= \tfrac{1}{2} = -B_1(0).

The Euler numbers are given by E_n=2^nE_n(\tfrac{1}{2}).

Explicit expressions for low degrees

The first few Bernoulli polynomials are:

\begin{align} 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{align}

The first few Euler polynomials are:

\begin{align} 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{align}

Maximum and minimum

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

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

M_n < \frac{2n!}{(2\pi)^n}

unless n is 2 modulo 4, in which case

M_n = \frac{2\zeta(n)n!}{(2\pi)^n}

(where \zeta(x) is the Riemann zeta function), while the minimum obeys

m_n > \frac{-2n!}{(2\pi)^n}

unless n is 0 modulo 4, in which case

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:

\Delta B_n(x) = B_n(x+1)-B_n(x)=nx^{n-1},\,
\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:

B_n'(x)=nB_{n-1}(x),\,
E_n'(x)=nE_{n-1}(x).\,

Translations

B_n(x+y)=\sum_{k=0}^n {n \choose k} B_k(x) y^{n-k}
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)^nB_n(x),\quad n \ge 0,
E_n(1-x)=(-1)^n E_n(x)\,
(-1)^n B_n(-x) = B_n(x) + nx^{n-1}\,
(-1)^n E_n(-x) = -E_n(x) + 2x^n\,
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,

where

[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) = -\frac{n!}{(2\pi i)^n}\sum_{k\not=0 }\frac{e^{2\pi ikx}}{k^n}= -2 n! \sum_{k=1}^{\infty} \frac{\cos\left(2 k \pi x- \frac{n \pi} 2 \right)}{(2 k \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) = -\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_\nu(x) = \sum_{k=0}^\infty \frac {\cos((2k+1)\pi x)} {(2k+1)^\nu}

and

S_\nu(x) = \sum_{k=0}^\infty \frac {\sin((2k+1)\pi x)} {(2k+1)^\nu}

for \nu > 1, the Euler polynomial has the Fourier series

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

and

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

Note that the C_\nu and S_\nu are odd and even, respectively:

C_\nu(x) = -C_\nu(1-x)

and

S_\nu(x) = S_\nu(1-x).

They are related to the Legendre chi function \chi_\nu as

C_\nu(x) = \mbox{Re} \chi_\nu (e^{ix})

and

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

x^n = \frac {1}{n+1} \sum_{k=0}^n {n+1 \choose k} B_k (x)

and

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 as

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) and

\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} = \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

\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(mx)= m^{n-1} \sum_{k=0}^{m-1} B_n \left(x+\frac{k}{m}\right)
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(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

\int_a^x B_n(t)\,dt = \frac{B_{n+1}(x)-B_{n+1}(a)}{n+1}
\int_a^x E_n(t)\,dt = \frac{E_{n+1}(x)-E_{n+1}(a)}{n+1}

Definite integrals

\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 \ge 1
\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.

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 :

\begin{align} &P_k(x) \text{ is continuous for all } k \neq 1 \\ &P_k'(x) \text{ exists and is continuous for } k=0, k \geq 3 \\ &P'_k(x) = kP_{k-1}(x), k \geq 3 \end{align}

See also

References

  1. ↑ D.H. Lehmer, "On the Maxima and Minima of Bernoulli Polynomials", American Mathematical Monthly, volume 47, pages 533–538 (1940)
  2. ↑ Zhi-Wei Sun; Hao Pan (2006). "Identities concerning Bernoulli and Euler polynomials". Acta Arithmetica 125: 21–39. arXiv:math/0409035. doi:10.4064/aa125-1-3.
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