The only issue worth checking is convergence, which follows from the fact that $|\alpha_{1}(\mu_{1},\mu_{2},\eta+n)-\frac{1}{4|n|\sqrt{a_{22}}}|\leq\frac{C(a_{ij},\mu_{1},\mu_{2},\eta)}{n^{2}}$, and likewise for $\alpha_{2},\alpha_{3}$. ∎