Editing Bernoulli polynomials (section)

==Fourier series==

The [[Fourier series]] of the Bernoulli polynomials is also a [[Dirichlet series]], given by the expansion
<math display="block">B_n(x) = -\frac{n!}{(2\pi i)^n}\sum_{k\not=0 }\frac{e^{2\pi ikx}}{k^n}= -2 n! \sum_{k=1}^{\infty} \frac{\cos\left(2 k \pi x- \frac{n \pi} 2 \right)}{(2 k \pi)^n}.</math>
Note the simple large ''n'' limit to suitably scaled trigonometric functions.

This is a special case of the analogous form for the [[Hurwitz zeta function]]
<math display="block">B_n(x) = -\Gamma(n+1) \sum_{k=1}^\infty \frac{ \exp (2\pi ikx) + e^{i\pi n} \exp (2\pi ik(1-x)) } { (2\pi ik)^n }. </math>

This expansion is valid only for {{math|0 ≤ ''x'' ≤ 1}} when {{math|''n'' ≥ 2}} and is valid for {{math|0 < ''x'' < 1}} when {{math|1=''n'' = 1}}.

The Fourier series of the Euler polynomials may also be calculated.  Defining the functions
<math display="block">\begin{align}
C_\nu(x) &= \sum_{k=0}^\infty \frac {\cos((2k+1)\pi x)} {(2k+1)^\nu} \\[3mu]
S_\nu(x) &= \sum_{k=0}^\infty \frac {\sin((2k+1)\pi x)} {(2k+1)^\nu}
\end{align}</math>
for <math>\nu > 1</math>, the Euler polynomial has the Fourier series
<math display="block">\begin{align}
C_{2n}(x) &= \frac{\left(-1\right)^n}{4(2n-1)!} \pi^{2n} E_{2n-1} (x) \\[1ex]
S_{2n+1}(x) &= \frac{\left(-1\right)^n}{4(2n)!} \pi^{2n+1} E_{2n} (x).
\end{align}</math>
Note that the <math>C_\nu</math> and <math>S_\nu</math> are odd and even, respectively:<math display="block">\begin{align}
C_\nu(x) &= -C_\nu(1-x) \\
S_\nu(x) &= S_\nu(1-x).
\end{align}</math>

They are related to the [[Legendre chi function]] <math>\chi_\nu</math> as
<math display="block">\begin{align}
C_\nu(x) &= \operatorname{Re} \chi_\nu (e^{ix}) \\
S_\nu(x) &= \operatorname{Im} \chi_\nu (e^{ix}).
\end{align}</math>