So $4$ days ago, I asked this question on here asking if there was any way that I would be able to evaluate an infinite product that I came up with. My strategy to solving said (doubly) infinite product was first noting that[$1$]$$\prod_{a\geq b}c^{d_a}\implies c^{\sum_{a\geq b}d_a}$$and then by noting that said infinite sum could be transformed into an integral (which is a strategy that can be done using Laplace Transform[$2$]) and then evaluating the integral and simplifying.
My question being: Say I wanted to evaluate my doubly infinite product[$3$] without first changing it into an infinte sum. Is there a way I would actually be able to do so, or is this the best method there is to evaluating my doubly infinite product already?
Notes
[$1$]You could either just read my justification for this method at the bottom of the original question of mine or just derive a justification on your own (If you need help: Think about the laws of exponents).
[$2$]You can look here for a proof or it's pretty easy to derive it on your own.
[$3$]For reference for anybody who isn't able/doesn't want to switch tabs, here (in my experience, it can be sort of annoying sometimes clicking on a link and then forgetting the reason why you clicked on the link) is the doubly infinite product that I originally evaluated (note that I am doing this to avoid possible confusion that might come up if I don't do so. This could also result in my question possibly being closed/deleted):$$\prod_{n\geq1}\prod_{k\geq1}e^{\frac{(-1)^{n+k}}{n^2k^2+3nk^2+3n^2k+2k^2+2n^2+9kn+6k+6n+4}}$$