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 the 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 order of the polynomials goes up. In the limit of large order, the Bernoulli polynomials, appropriately scaled, approach the sine and cosine functions.
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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>
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
These polynomial sequences are Appel sequences:
- <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 sequences. (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>
Multiplication theorems
- <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)<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)<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.)
- Tom M. Apostol Introduction to Analytic Number Theory, (1976) Springer-Verlag, New York. (See Chapter 12.11)