Letting the lattice $\Lambda := \{w_{mn} := m + in : m,n \in \mathbb{Z} \}$, let $H(z) := z \prod_{w \in \Lambda } (1 - \frac{z^4}{w^4})$. I am trying to show that $H(z-1) = -e^{-\pi(z-1)/2}H(z)$, following Exercise 8 in Section 2.5 of Taylor's Intro. to Complex Analysis (about infinite products and Weierstrauss factorization theorem).
Here's my idea so far (following the hint): we have $H(z) = \lim_{k \rightarrow \infty} H_k(z)$, where $H_k(z) = \alpha_k \prod_{|m|,|n| \leq k} (z - w_{mn})$ with $\alpha_k := \prod_{|m|,|n| \leq k, m + in \ne 0} (- w_{mn})^{-1}$.
Then, $H_k(z-1) = \alpha_k \prod_{|n|, |m| \leq k } ((z-1)-w_{mn}) = \alpha_k \prod_{0 \leq |n|, |m|-1 \leq k } (z-w_{mn})$.
Hence, $\frac{H_k(z-1)}{H_k(z)} = \frac{\prod_{|n| \leq k} (z-w_{k+1,n})}{\prod_{|n| \leq k } (z-w_{0,n})} = \prod_{|n| \leq k} \frac{z-k-1-in}{z-in} = \prod_{|n| \leq k} (1 - \frac{k+1}{z-in})$.
I am not sure how to proceed from here. In particular, I'm unsure if I'm correct till this step even, since I want to show that the final product above is $-e^{-\pi(z-1)/2}$: but the above is undefined at $z = in$, which is of course not true for this exponential. I'd appreciate any help!
It might be helpful to use $\sum_{l = -k}^k \frac{1}{k+il} \rightarrow \pi/2$ as $k \rightarrow \infty$.