Utilisateur:Wiz/Brouillon/Polynômes de Bernoulli

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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 due to the fact that they are Sheffer sequences 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.

Sommaire

1 Generating functions
2 Characterization by a differential operator
3 Explicit formula
4 The Bernoulli and Euler numbers
5 Explicit expressions for low degrees
6 Differences
7 Derivatives
8 Translations
9 Symmetries
10 Fourier series
11 Inversion
12 Relation to falling factorial
13 Multiplication theorems
14 Integrals
15 References

[modifier] 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

$\frac{2 e^{xt}}{e^t+1}= \sum_{n=0}^\infty E_n(x) \frac{t^n}{n!}.$

[modifier] Characterization 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.

[modifier] 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 ζ(1 - n, x)

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

An explicit formula for the Euler polynomials is given by

$E_m(x)= \sum_{n=0}^m \frac{1}{2^n} \sum_{k=0}^n (-1)^k {n \choose k} (x+k)^m.$

[modifier] The Bernoulli and Euler numbers

The Bernoulli numbers are given by $B n = B n (0).$

The Euler numbers are given by $E n = 2 n E n (1 / 2).$

[modifier] 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

$E_0(x)=1\,$

$E_1(x)=x-1/2\,$

$E_2(x)=x^2-x\,$

$E_3(x)=x^3-\frac{3}{2}x^2+\frac{1}{4}\,$

$E_4(x)=x^4-2x^3+x\,$

$E_5(x)=x^5-\frac{5}{2}x^4+\frac{5}{2}x^2-\frac{1}{2}\,$

$E_6(x)=x^6-3x^5+5x^3-3x.\,$

[modifier] Differences

The Bernoulli and Euler polynomials obey many relations from umbral calculus:

B n (x + 1) - B n (x) = n x n - 1

E n (x + 1) + E n (x) = 2 x n .

[modifier] Derivatives

These polynomial sequences are Appel sequences:

B n'(x) = n B n - 1 (x)

E n'(x) = n E n - 1 (x).

[modifier] 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 Appel sequences. (Hermite polynomials are another example.)

[modifier] Symmetries

B n (1 - x) = ( - ) n B n (x)

E n (1 - x) = ( - ) n E n (x)

( - ) n B n ( - x) = B n (x) + n x n - 1

( - ) n E n ( - x) = - E n (x) + 2 x n

[modifier] Fourier series

The Fourier series of the Bernoulli polynomials is also a Dirichlet series and is a special case of the Hurwitz zeta function

$B_n(x) = -\Gamma(n+1) \sum_{k=1}^\infty \frac{ \exp (2\pi ikx) + \exp (2\pi ik(1-x)) } { (2\pi ik)^n }.$

[modifier] Inversion

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

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

[modifier] 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.

[modifier] Multiplication theorems

The multipliction theorems were given by Joeseph Ludwig Raabe in 1851:

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

$E_n(mx)= m^n \sum_{k=0}^{m-1} (-1)^k E_n \left(x+\frac{k}{m}\right) \quad \mbox{ for } m=1,3,...$

$E_n(mx)= \frac{-2}{n+1} m^n \sum_{k=0}^{m-1} (-1)^k B_{n+1} \left(x+\frac{k}{m}\right) \quad \mbox{ for } m=2,4,...$

[modifier] 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

$\int_0^1 B_n(t) B_m(t)\,dt = (-1)^{n-1} \frac{m! n!}{(m+n)!} B_{n+m} \quad \mbox { for } m,n \ge 1$

$\int_0^1 E_n(t) E_m(t)\,dt = (-1)^{n} 4 (2^{m+n+2}-1)\frac{m! n!}{(m+n+2)!} B_{n+m+2}$

[modifier] References

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

Tom M. Apostol Introduction to Analytic Number Theory, (1976) Springer-Verlag, New York. (See Chapter 12.11)

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.)

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 due to the fact that they are [[Sheffer sequence]]s 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 [[trigonometric function|sine and cosine functions]]. ==Generating functions== The [[generating function]] for the Bernoulli polynomials is :<math>\frac{t e^{xt}}{e^t-1}= \sum_{n=0}^\infty B_n(x) \frac{t^n}{n!}.</math> The generating function for the Euler polynomials is :<math>\frac{2 e^{xt}}{e^t+1}= \sum_{n=0}^\infty E_n(x) \frac{t^n}{n!}.</math> ==Characterization by a differential operator== The Bernoulli polynomials are also given by :<math>B_n(x)={D \over e^D -1} x^n</math> where ''D'' = ''d''/''dx'' is differentiation with respect to ''x'' and the fraction is expanded as a [[formal power series]]. ==Explicit formula== An explicit formula for the Bernoulli polynomials is given by :<math>B_m(x)= \sum_{n=0}^m \frac{1}{n+1} \sum_{k=0}^n (-1)^k {n \choose k} (x+k)^m.</math> Note the remarkable similarity to the globally convergent series expression for the [[Hurwitz zeta function]]. Indeed, one has :<math>B_n(x) = -n \zeta(1-n,x)</math> where <math>\zeta(s,q)</math> is the Hurwitz zeta; this, in a certain sense, the Hurwitz zeta generalizes the Bernoulli polynomials to non-integer values of ''n''. An explicit formula for the Euler polynomials is given by :<math>E_m(x)= \sum_{n=0}^m \frac{1}{2^n} \sum_{k=0}^n (-1)^k {n \choose k} (x+k)^m.</math> ==The Bernoulli and Euler numbers== The [[Bernoulli number]]s are given by <math>B_n=B_n(0).</math> The [[Euler number]]s are given by <math>E_n=2^nE_n(1/2).</math> ==Explicit expressions for low degrees== The first few Bernoulli polynomials are: :<math>B_0(x)=1\,</math> :<math>B_1(x)=x-1/2\,</math> :<math>B_2(x)=x^2-x+1/6\,</math> :<math>B_3(x)=x^3-\frac{3}{2}x^2+\frac{1}{2}x\,</math> :<math>B_4(x)=x^4-2x^3+x^2-\frac{1}{30}\,</math> :<math>B_5(x)=x^5-\frac{5}{2}x^4+\frac{5}{3}x^3-\frac{1}{6}x\,</math> :<math>B_6(x)=x^6-3x^5+\frac{5}{2}x^4-\frac{1}{2}x^2+\frac{1}{42}.\,</math> The first few Euler polynomials are :<math>E_0(x)=1\,</math> :<math>E_1(x)=x-1/2\,</math> :<math>E_2(x)=x^2-x\,</math> :<math>E_3(x)=x^3-\frac{3}{2}x^2+\frac{1}{4}\,</math> :<math>E_4(x)=x^4-2x^3+x\,</math> :<math>E_5(x)=x^5-\frac{5}{2}x^4+\frac{5}{2}x^2-\frac{1}{2}\,</math> :<math>E_6(x)=x^6-3x^5+5x^3-3x.\,</math> ==Differences== The Bernoulli and Euler polynomials obey many relations from [[umbral calculus]]: :<math>B_n(x+1)-B_n(x)=nx^{n-1}</math> :<math>E_n(x+1)+E_n(x)=2x^n.</math> ==Derivatives== These [[polynomial sequence]]s are [[Appel sequence]]s: :<math>B_n'(x)=nB_{n-1}(x)</math> :<math>E_n'(x)=nE_{n-1}(x).</math> ==Translations== :<math>B_n(x+y)=\sum_{k=0}^n {n \choose k} B_k(x) y^{n-k}</math> :<math>E_n(x+y)=\sum_{k=0}^n {n \choose k} E_k(x) y^{n-k}</math> These identities are also equivalent to saying that these polynomial sequences are [[Appel sequence]]s. ([[Hermite polynomials]] are another example.) ==Symmetries== :<math>B_n(1-x)=(-)^n B_n(x)</math> :<math>E_n(1-x)=(-)^n E_n(x)</math> :<math>(-)^n B_n(-x) = B_n(x) + nx^{n-1}</math> :<math>(-)^n E_n(-x) = -E_n(x) + 2x^n</math> ==Fourier series== The [[Fourier series]] of the Bernoulli polynomials is also a [[Dirichlet series]] and is a special case of the [[Hurwitz zeta function]] :<math>B_n(x) = -\Gamma(n+1) \sum_{k=1}^\infty \frac{ \exp (2\pi ikx) + \exp (2\pi ik(1-x)) } { (2\pi ik)^n }. </math> ==Inversion== The Bernoulli polynomials may be inverted to express the [[monomial]] in terms of the polynomials. Specifically, one has :<math>x^n = \frac {1}{n+1} \sum_{k=0}^n {n+1 \choose k} B_k (x). </math> ==Relation to falling factorial== The Bernoulli polynomials may be expanded in terms of the [[falling factorial]] <math>(x)_k</math> as :<math>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} </math> where <math>B_n=B_n(0)</math> and :<math>\left\{ \begin{matrix} n \\ k \end{matrix} \right\} = S(n,k)</math> denotes the [[Stirling number of the second kind]]. The above may be inverted to express the falling factorial in terms of the Bernoulli polynomials: :<math>(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) </math> where :<math>\left[ \begin{matrix} n \\ k \end{matrix} \right] = s(n,k)</math> denotes the [[Stirling number of the first kind]]. ==Multiplication theorems== The [[multipliction theorem]]s were given by [[Joeseph Ludwig Raabe]] in [[1851]]: :<math>B_n(mx)= m^{n-1} \sum_{k=0}^{m-1} B_n \left(x+\frac{k}{m}\right)</math> :<math>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,...</math> :<math>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,...</math> ==Integrals== Indefinite integrals :<math>\int_a^x B_n(t)\,dt = \frac{B_{n+1}(x)-B_{n+1}(a)}{n+1}</math> :<math>\int_a^x E_n(t)\,dt = \frac{E_{n+1}(x)-E_{n+1}(a)}{n+1}</math> Definite integrals :<math>\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 </math> :<math>\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}</math> ==References== * Milton Abramowitz and Irene A. Stegun, eds. ''[[Handbook of Mathematical Functions]] with Formulas, Graphs, and Mathematical Tables'', (1972) Dover, New York. ''(See [http://www.math.sfu.ca/~cbm/aands/page_804.htm Chapter 23])'' * Tom M. Apostol ''Introduction to Analytic Number Theory'', (1976) Springer-Verlag, New York. ''(See Chapter 12.11)'' * Jesus Guillera and Jonathan Sondow, ''[http://arxiv.org/abs/math.NT/0506319 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.)'' ---- [[Category:Théorie des nombres]] [[Category:Pôlynome remarquable]] [[it:Polinomi di Bernoulli]]