We start with the initial step by assuming $\alpha-\beta>1$ and $\beta>1$. Then Lemma A.3 in this case shows that the desired asymptotics hold in this case.

Applying the recurrence formula (A.33), we can extend the domain of indices to $\alpha-\beta>0$ and $\beta>1$. Then applying (A.32), one can obtain the desired asymptotics for all $\beta\in\mathbb{R}$. The proof is done. ∎