Bernoulli polynomials

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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. Image:Bernoulli polynomials.svg

Contents

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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>
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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.

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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.

The inner sum may be understood to be the n 'th forward difference of <math>x^m</math>; that is,

<math>\Delta^n x^m = \sum_{k=0}^n (-1)^{n-k} {n \choose k} (x+k)^m</math>

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

<math>B_m(x)= \sum_{n=0}^m \frac{(-1)^n}{n+1} \Delta^n x^m </math>.

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

<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>

This may also be written in terms of the Euler numbers <math>E_k</math> as

<math>E_m(x)=

\sum_{k=0}^m {m \choose k} \left(\frac{1}{2}\right)^k \left(x-\frac{1}{2}\right)^{m-k} E_k\,.</math>

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The Bernoulli and Euler numbers

The Bernoulli numbers are given by <math>B_n=B_n(0).</math>

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

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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>
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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>
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Derivatives

These polynomial sequences are Appel sequences:

<math>B_n'(x)=nB_{n-1}(x)</math>
<math>E_n'(x)=nE_{n-1}(x).</math>
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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 sequences. (Hermite polynomials are another example.)

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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>
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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) + (-1)^n \exp (2\pi ik(1-x)) } { (2\pi ik)^n }. </math>

This expansion is valid only for <math>0\leq x\leq 1</math> when <math>n \geq 2</math> and is valid for <math>0< x< 1</math> when <math>n = 1</math>.

Defining the functions

<math>C_\nu(x) = \sum_{k=0}^\infty

\frac {\cos((2k+1)\pi x)} {(2k+1)^\nu}</math> and

<math>S_\nu(x) = \sum_{k=0}^\infty

\frac {\sin((2k+1)\pi x)} {(2k+1)^\nu}</math>

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

<math>C_{2n}(x) = \frac{(-1)^n}{4(2n-1)!}

\pi^{2n} E_{2n-1} (x)</math>

and

<math>S_{2n+1}(x) = \frac{(-1)^n}{4(2n)!}

\pi^{2n+1} E_{2n} (x)</math>

Note that the <math>C_\nu</math> and <math>S_\nu</math> are odd and even, respectively:

<math>C_\nu(x) = -C_\nu(1-x)</math>

and

<math>S_\nu(x) = S_\nu(1-x)</math>

They are related to the Legendre chi function <math>\chi_\nu</math> as

<math>C_\nu(x) = \mbox{Re} \chi_\nu (e^{ix})</math>

and

<math>S_\nu(x) = \mbox{Im} \chi_\nu (e^{ix})</math>
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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>

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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.

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Multiplication theorems

The multipliction theorems 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>

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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>

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References

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

it:Polinomio di Bernoulli

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