Is it possible to obtain the probability mass function for the discrete random variable $X$ associated with the following probability generating function?
$$ \mathbb E[s^X]=\left(\frac{1-\delta P}{1-sP}\right)^{\frac{1-s}{\delta-s}} $$
In the limit $\delta\to1$, this is the generating function of the Geometric($P$) distribution, which models the number of successes before the first failure given a success probability $P$. It would be great to obtain the distribution of $X$ for any $0<\delta<1$, or at least some sort of asymptotic expansion about $\delta\approx 1$. Any ideas would be very much appreciated!