Bernoulli polynomials - Definition

In mathematics, the Bernoulli polynomials occur in the study of many special functions and in particular the Riemann zeta function.

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>

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>

Explicit expressions for low orders

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

<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>

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>

References

• M. Abramowitz and I. A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, (1972) Dover, New York. (See Chapter 23.); wiki: Abramowitz and Stegun.

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

• Linas Vepstas, The Bernoulli Operator, the Gauss-Kuzmin-Wirsing Operator, and the Riemann Zeta (http://www.linas.org/math/chap-gkw/gkw.html)