Bernoulli polynomials

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.


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_k x^{n-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.

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 operator

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

on polynomials "f", is the same as

:egin{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}

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 zeta(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 "n"th 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 is equal to

:Delta = e^D - 1,

where "D" is differentiation with respect to "x", we have

:{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 "m"th-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}left(frac{1}{2} ight)^k left(x-frac{1}{2} ight)^{m-k} E_k,.

ums of "p"th powers

We have

:sum_{k=0}^{x} k^p = frac{B_{p+1}(x+1)-B_{p+1}(0)}{p+1}.

See Faulhaber's formula for more on this.

The Bernoulli and Euler numbers

The Bernoulli numbers are given by B_n=B_n(0).

The Euler numbers are given by E_n=2^nE_n(1/2).

Explicit expressions for low degrees

The first few Bernoulli polynomials are::B_0(x)=1,:B_1(x)=x-1/2,:B_2(x)=x^2-x+1/6,:B_3(x)=x^3-frac{3}{2}x^2+frac{1}{2}x,:B_4(x)=x^4-2x^3+x^2-frac{1}{30},:B_5(x)=x^5-frac{5}{2}x^4+frac{5}{3}x^3-frac{1}{6}x,: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:E_0(x)=1,:E_1(x)=x-1/2,:E_2(x)=x^2-x,:E_3(x)=x^3-frac{3}{2}x^2+frac{1}{4},:E_4(x)=x^4-2x^3+x,:E_5(x)=x^5-frac{5}{2}x^4+frac{5}{2}x^2-frac{1}{2},:E_6(x)=x^6-3x^5+5x^3-3x.,

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)=2x^n.

(Δ is the forward difference operator).

These polynomial sequences are Appell sequences:




: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.)


: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,

Zhi-Wei Sun and Hao Pan [] established the following surprising symmetric 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,


: [s,t;x,y] _n=sum_{k=0}^n(-1)^k{s choose k}{tchoose {n-kB_{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!}{(2pi i)^n}sum_{k ot=0 }frac{e^{2pi i x{k^n}.

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 (2pi ikx) + e^{ipi n} exp (2pi ik(1-x)) } { (2pi ik)^n }.

This expansion is valid only for 0 &le; "x" &le; 1 when "n" &ge; 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_ u(x) = sum_{k=0}^infty frac {cos((2k+1)pi x)} {(2k+1)^ u}


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

for u > 1, the Euler polynomial has the Fourier series

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


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

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

:C_ u(x) = -C_ u(1-x)


:S_ u(x) = S_ u(1-x).

They are related to the Legendre chi function chi_ u as

:C_ u(x) = mbox{Re} chi_ u (e^{ix})


:S_ u(x) = mbox{Im} chi_ u (e^{ix}).


The Bernoulli polynomials may be inverted to express the monomial in terms of the polynomials. Specifically, one has

:x^n = frac {1}{n+1} sum_{k=0}^n {n+1 choose k} B_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}^nfrac{n+1}{k+1}left{ egin{matrix} n \ k end{matrix} ight}(x)_{k+1} where B_n=B_n(0) and

:left{ egin{matrix} n \ k end{matrix} ight} = 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 [ egin{matrix} n \ k end{matrix} ight] left(B_{k+1}(x) - B_{k+1} ight)

where :left [ egin{matrix} n \ k end{matrix} ight] = s(n,k)

denotes the Stirling number of the first kind.

Multiplication theorems

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

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

:E_n(mx)= m^n sum_{k=0}^{m-1} (-1)^k E_n left(x+frac{k}{m} ight)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} ight)quad mbox{ for } m=2,4,dots


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 evalated 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.


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

* See chapter 12.11 of Apostol IANT

* Djurdje Cvijović and Jacek Klinowski, "New formulae for the Bernoulli and Euler polynomials at rational arguments", Proceedings of the American Mathematical Society123"' (1995), 1527-1535.

* Jesus Guillera and Jonathan Sondow, " [ Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent] " (2005) "(Reviews relationship to the Hurwitz zeta function and Lerch transcendent.)"

* [ Kurtulan, Ali Burak "Bernoulli Polynomials and Their Applications"]


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Bernoulli — can refer to: *any one or more of the Bernoulli family of Swiss mathematicians in the eighteenth century, including: ** Daniel Bernoulli (1700–1782), developer of Bernoulli s principle ** Jakob Bernoulli (1654–1705), also known as Jean or Jacques …   Wikipedia

  • Bernoulli number — In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers with deep connections to number theory. They are closely related to the values of the Riemann zeta function at negative integers. There are several conventions for… …   Wikipedia

  • Bernoulli process — In probability and statistics, a Bernoulli processis a discrete time stochastic process consisting ofa sequence of independent random variables taking values over two symbols. Prosaically, a Bernoulli process is coin flipping, possibly with an… …   Wikipedia

  • Bernoulli family — The Bernoullis were a family of traders and scholars from Basel, Switzerland. The founder of the family, Leon Bernoulli, immigrated to Basel from Antwerp in the Flanders in the 16th century.The Bernoulli family has produced many notable artists… …   Wikipedia

  • Euler–Worpitzky–Chen polynomials — Introduction = The Euler Worpitzky Chen polynomials are closely related to the family of Euler Bernoulli polynomials and numbers. The coefficients of the polynomialsare integers, in contrast to the coefficients of the Euler and Bernoulli… …   Wikipedia

  • Euler–Maclaurin formula — In mathematics, the Euler–Maclaurin formula provides a powerful connection between integrals (see calculus) and sums. It can be used to approximate integrals by finite sums, or conversely to evaluate finite sums and infinite series using… …   Wikipedia

  • Faulhaber's formula — In mathematics, Faulhaber s formula, named after Johann Faulhaber, expresses the sum:sum {k=1}^n k^p = 1^p + 2^p + 3^p + cdots + n^pas a ( p + 1)th degree polynomial function of n , the coefficients involving Bernoulli numbers.Note: By the most… …   Wikipedia

  • Multiplication theorem — In mathematics, the multiplication theorem is a certain type of identity obeyed by many special functions related to the gamma function. For the explicit case of the gamma function, the identity is a product of values; thus the name. The various… …   Wikipedia

  • Polylogarithm — Not to be confused with polylogarithmic. In mathematics, the polylogarithm (also known as Jonquière s function) is a special function Lis(z) that is defined by the infinite sum, or power series: It is in general not an elementary function, unlike …   Wikipedia

  • List of mathematics articles (B) — NOTOC B B spline B* algebra B* search algorithm B,C,K,W system BA model Ba space Babuška Lax Milgram theorem Baby Monster group Baby step giant step Babylonian mathematics Babylonian numerals Bach tensor Bach s algorithm Bachmann–Howard ordinal… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.