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I know the exponential generating function for the Bernoulli polynomial $B_n(x)$:$$\frac{te^{tx}}{e^t-1}=\sum_{n=0}^\infty B_n(x)\frac{t^n}{n!}.$$ But is there an ordinary generating function? i.e a function $f$ such that $$f(t,x)=\sum_{n=0}^\infty B_n(x)t^n$$ valid for some open interval of $x$ and some open interval of $t$.

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No. The exponential generating function has a pole at $t = 2 \pi i$, so for any $x$, $B_n(x)$ grows at least as fast as about $\frac{n!}{(2\pi)^n}$, hence the ordinary generating function has zero radius of convergence in $t$.

When possible, you can convert exponential generating functions to ordinary ones using a variant of the Laplace transform.

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