# Special infinite product

I would need help or references to in order to calculate the following infinite product: $$\lim_{N\rightarrow \infty}\prod_{i=-N}^N\dfrac{\sqrt{1-\left(\dfrac{c_i}{\dfrac{d^{(i)}x}{dt^{(i)}}}\right)^2}}{\sqrt{1-\left(\dfrac{\dfrac{d^{(i)}x}{dt^{(i)}}}{C_i}\right)^2}}$$ and where $C_i$ and $c_i$ are some "constants". If we set "natural units" with $C_i=c_i=1$ for simplicity, and we write $\dfrac{d^{(i)}x}{dt^{(i)}}\equiv q^i$, we get something like

$$\prod_{i=-\infty}^\infty\dfrac{\sqrt{1-\left(\dfrac{1}{q^i}\right)^2}}{\sqrt{1-\left(q^i\right)^2}}=\prod_{i=-\infty}^\infty\dfrac{\sqrt{-1}}{q^i}$$

Can this last product be regularized in a consistent way, e.g. in the zeta regularization procedure?Does convergence imply some conditions on $C_i,c_i$ if I don't set them to the unit?

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Define "renormalized in a consistent way". – Did Jul 4 '13 at 17:32
In the above expression, I suppose you noticed that including negative derivatives (as integrals, of course), introduce a infinite number of integration constants. So, how could I regularize all this? I meant regularized, sorry. I corrected that mistake. – riemannium Jul 5 '13 at 10:32
I suppose you noticed... You suppose correctly. So, I guess that now, to make this "A Real Question", we need a definition of "regularization". – Did Jul 5 '13 at 10:38
For instance, would zeta function regularization be valid? I also wonder if my product could be related to some q-gamma functions for 0<q<1 and or some other q-function. I am not an expert (yet) in those fields, but I really really need to know how could I regularize this product. I am not too worried about having different options, but I would prefer a zeta function approach and/or a modular q-function identity that could be matched with this product. – riemannium Jul 5 '13 at 10:42