In William Feller's 1st book p.272
It said the generating function $\Phi$ satisfies
\begin{equation*} qs\Phi^2(s) - \Phi(s) + ps = 0 \end{equation*}
so it has two roots. The first root is unbounded near $s = 0$.
So the generating function is given by the unique bounded solution
\begin{equation*} \Phi(s) = \frac{1 - \sqrt{1 - 4pqs^2}}{2qs}. \end{equation*}
Also, \begin{equation*} \Phi(s) = \sum_{n=0}^\infty \phi_ns^n. \end{equation*}
What I don't understand is how the coefficients \begin{equation*} \phi_{2k-1} = \frac{(-1)^{k-1}}{2q} \binom{1/2}{k} (4pq)^k, \quad \phi_{2k} = 0 \end{equation*} come up with binomial expansion of the unique root.
To be clear, I want to know how the unique root of $\Phi$ is converted to the form $(1 + t)^a$ and how and why the new form have the coefficients above?