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:
 * $\ds \frac {t e^{t x} } {e^t - 1} = \sum_{k \mathop = 0}^\infty \frac {\map {B_k} x} {k!} t^k$

Proof
By definition of the generating function for Bernoulli numbers:


 * $\ds \frac t {e^t - 1} = \sum_{k \mathop = 0}^\infty \frac {B_k} {k!} t^k$

By Power Series Expansion for Exponential Function:


 * $\ds e^{t x} = \sum_{k \mathop = 0}^\infty \frac {x^k} {k!} t^k$

Thus:


 * $\ds \frac {t e^{t x} } {e^t - 1} = \paren {\sum_{k \mathop = 0}^\infty \frac {B_k} {k!} t^k} \paren {\sum_{k \mathop = 0}^\infty \frac {x^k} {k!} t^k}$

Combining like powers of $t$ we obtain: