I think I found out.
$$\displaystyle (1-\frac{1}{p^{s+1}-\frac{1}{p^{t+1}+\frac{1}{p^{t+s+1}})^k=\sum_{\alpha_1,...,\aplha_4}\frac{k!}{\alpha_1!,...\alpha_4!}1^{\alpha_1}p^{^-(s+1)\alpha_2}p^{-(t+1)\alpha_3}p^{-(s+t+1)\alpha_4} $$
Right here we get $ \displaystyle (1-\frac{1}{p^{s+1}-\frac{1}{p^{t+1}+\frac{1}{p^{t+s+1}}) and rest of it is $O(1/p^2).
Since
$$ \displaystyle \frac{(1-p^-{s+1})(1-p^-{t+1}){(1-p^-{2+s+t})}={(1-p^-{s+1})(1-p^-{t+1})(1+p^-{2+s+t}+O(1/p^2). $$
equation holds.