I'm trying to derive the probability generating function (pgf) for the logarithmic series distribution, and not getting the expected form $\frac{\log{(1-qs)}}{\log{(1-q)}}$.
It seems that pgfs are usually written for discrete distributions on $k\geq 0$, whereas for this distribution the support is $k\geq1$ in the form usually written. So I parameterise the distribution as follows to get to the support $k\geq0$:
$$p(k) = \frac{-1}{\log{(1-q)}} \frac{q^{k+1}}{k+1}, k \geq0.$$
So then to find the pgf, I have
$$G_K(s) = \sum_{k=0}^\inf{p(k) s^k}$$
$$G_K(s) = \frac{-1}{\log{(1-q)}} \sum_{k=0}^\inf{\frac{q^{k+1}}{k+1} s^k}$$
which I can reparametrise "back" to get
$$G_K(s) = \frac{-1}{\log{(1-q)}} \sum_{k=1}^\inf{\frac{q^{k}}{k} s^{k-1}}.$$
By rearranging and using the fact that $\sum_{n=1}^{\inf}{x^n / n} = -\log{(1-x)}$, I end up with
$$G_K(s) = \frac{\log{(1-qs)}}{s \log{(1-q)}}$$
This doesn't match what I see elsewhere, and it's merely the presence of that $\frac{1}{s}$ factor that is the problem, which clearly comes directly from my reparametrisation. Have I derived the correct pgf, or have I misunderstood how to handle the support of a probability distribution when constructing the pgf? Where is my error?