Problem Statement:
I have a function of the form
$$ F(z) = G(z)^a H(z)^b$$ where $ G(z) = \sqrt{1-Qz} + \sqrt{1-Q},$ $ H(z) = \frac{1-\sqrt{1-Qz}}{z}$, and $a$, $b$, $Q$ are non-integer constants. This function arises as a probability generating function for a random variable, so I need to compute
$$ p(n) = \frac{1}{n!}F^{(n)}(0+)$$ to determine the probability mass function $p(n)$. How can I best compute $F^{(n)}(z)$?
Ideas:
One idea is to figure out $G^{(n)}(z)$ and $H^{(n)}(z)$, then figure out how recursive application of the product rule leads to an expression of $F^{(n)}(z)$ in terms of these.
So far I have the general Leibniz rule $$ [f(z)g(z)]^{(n)} = \sum_{k=0}^n {n \choose k} f^{(n-k)}(z)g^{(k)}(z),$$ and the derivatives $$ G^{(n)} (z) = - \big(\frac{Q}{2}\big)^n (2n-1)!! (1-Qz)^{-(2n-1)/2}. $$ I'll update as I make more progress...
Any pointers for me to solve this repeated differentiation problem?
Then $$(1+Qz)^{a/2} = \sum_{k \ge 0} {a/2 \choose m} Q^m z^m$$ where ${a/2 \choose m} = \prod_{j=1}^m \frac{a/2-(j-1)}{j}$ and $$(fg)^{(k)}= \sum_{l=0}^k {k \choose l} f^{(l)} g^{(k-l)}$$