Bernoulli polynomials

In mathematics, the Bernoulli polynomials, named after Jacob Bernoulli, 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.

Representations

The Bernoulli polynomials B_n 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 b_k 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 x^m; 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 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_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 E_k 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 $E_{n}=2^{n}E_{n}({\tfrac {1}{2}}).$

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 B_n(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 B_n(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 $m \geq 1$ ,

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 $P n (x)$ 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.

The following properties are of interest, valid for all $x$ :

{\begin{aligned}&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{aligned}}

References

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

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., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, 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: 1527–1535. doi:10.2307/2161144.
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. (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. 97. Cambridge: Cambridge Univ. Press. pp. 495–519. ISBN 0-521-84903-9.

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Bernoulli polynomials

Representations

Explicit formula

Generating functions

Representation by a differential operator

Representation by an integral operator

Another explicit formula

Sums of pth powers

The Bernoulli and Euler numbers

Explicit expressions for low degrees

Maximum and minimum

Differences and derivatives

Translations

Symmetries

Fourier series

Inversion

Relation to falling factorial

Multiplication theorems

Integrals

Periodic Bernoulli polynomials

See also

References