Lower bound the ratio of the combinatorial quantities

Question

Suppose $p < q$, $s = p^{d}$ for some fixed $d \in (0,1)$, let $p$ goes to infinity, define the following quantity, \begin{aligned} \quad f(j) = \sum_{i = 0}^{\min(j,s)}{s \choose i}{p-s \choose i}{s \choose j-i}{q-s \choose j-i} \end{aligned} How to lower bound $\frac{f(j+1)}{f(j)}$? That is \begin{aligned} \frac{f(j+1)}{f(j)} \geq \;? \end{aligned}