Generating Function of Bernoulli Polynomials

Theorem
Let $\map {B_n} x$ denote the $n$th Bernoulli polynomial.

Then the generating function for $B_n$ is:
 * $\displaystyle \frac {t e^{t x} } {e^t - 1} = \sum_{k \mathop = 0}^\infty \frac {\map {B_k} x} {k!} t^k$

Proof
By definition of the generating function for Bernoulli numbers:

By Power Series Expansion for Exponential Function:

Thus:

Combining like powers of $t$ we obtain: