Bell series of a quotient of arithmetic functions?

If $f,g$ are multiplicative functions (with bell series $F_p(x), G_p(x)$) so is $n \mapsto f(n)/g(n)$, what is its Bell series? (or is there no nice way to write it in terms of them?)

I think it's not possible since it's composition in the case of completely multiplicative functions. I just want to know bell series for $\frac{n}{\varphi(n)}$ and $\sum_{d|n} \frac{\mu(d)^2}{\varphi(d)}$.

For the first I got $1 + \frac{p}{p-1}\cdot\frac{1}{1-x}$ and for the second $\tfrac{1}{1-x}\left(1 + \frac{1}{p-1}\right)$ so again I have this wrong.

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You do not need a formula for your problem, just calculate the Bell series of the two expressions. In the first case, you get a geometric series, in the second case a finite sum.

For $\frac n {\varphi (n)}$: $$1+x \frac p {p-1} + x^2\frac {p^2}{p(p-1)}+\dots = 1+\frac{xp}{p-1} \cdot (1+x+x^2+\dots)= 1+\frac{xp}{(p-1)(1-x)}$$

For $\frac {\mu(n)^2} {\varphi (n)}$ : $$1+x \frac 1 {p-1}$$

Now, to get your original identity, you needed to compose the second expression with the function that is identical one, so multiplying with $\frac 1 {1-x}$.

Note that for the general question, there is no hope for a simple answer as this is related to the Hadamard product and the extraction of the "diagonal" of a two-variable power-series. compare: Formal power series coefficient multiplication

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Thank you! ----- –  quanta Apr 23 '11 at 16:51