Composition of Formal Power Series

Question

Is there a known closed form for the composition of the following power series: $$f(x) = \exp(x) = \sum_k \frac{x^k}{k!}$$ $$g(x) = \frac{1}{1-x} = \sum_n x^n$$

I'd like a closed form power series for $g(f(x)) = \frac{1}{1-\exp (x)}$. I've been trying to use the di Bruno formula from Wikipedia but I find it very confusing, I'd appreciate some help.

Answer 1

I assume you mean a closed form for the coefficients. There is no simple closed form known (or at least that I know of), but these are essentially the Bernoulli numbers defined by $$ \frac{x}{e^x - 1} = \sum_{n=0}^\infty B_n \frac{x^n}{n!}.$$ Your sum is then $$\frac{1}{1-e^{x}} = \frac{-1}{x} \frac{x}{e^x-1} = -\sum_{n=0}^\infty B_n \frac{x^{n-1}}{n!}.$$

Note that there is an $x^{-1}$ term here, so this not actually a power series, but a Laurent series.

Answer 2

hint

$$g(f(x))=\frac{1}{x}\frac{1}{\frac{1-e^x}{x}}=$$ $$\frac{1}{x}g(h(x))$$ with $$h(x)=\frac{e^x-1}{x}-1$$