Let $1<\alpha<\beta<3/2$. Set
$$
S(n)= \sum_{i\neq j,~ i,j>0} [i^\alpha+j^\beta]^{-1}[(i+n)^\alpha+(j+n)^\beta]^{-1}.
$$
One can check that $S(n)$ is finite. My question is when $n\rightarrow \infty$, what is the asymptotics of $S(n)$, e.g., if it is asymptotically a power function? If yes, what is the exponent? 

Remark:
When $\alpha=\beta$, this problem can be resolved using an integral approximation argument (rewriting the sum as a double integral by replacing $\frac{i}{n}$ with $\frac{[nx]+1}{n}$, $\frac{j}{n}$ with $\frac{[ny]+1}{n}$  and let $n\rightarrow\infty$ through the Dominated Convergence Theorem) which yields $S(n)\sim c n^{2\alpha-2}$ for some $c>0$. But when $\alpha<\beta$, the similar argument seems hard to apply.