$$\lim_{\omega \to \infty} \prod_{N=1}^{\omega} {{1+e^{b \cdot c^{-N}}} \over 2}$$
For instance, the Lerch Transcendent is a analogous example of a special function that defines the sum of a useful infinite series. If it helps,
$$ \prod_{N=1}^{\infty} {{1+e^{b \cdot c^{-N}}} \over 2} \sim \left({{e^b-1} \over b} \right)^{1/(c-1)}$$
This asymptotic is exact if $c=2$. Here's a non-trivial application of the above.
I'd like the name of this infinite product so that I can learn more about it. For instance, does it have a closed form? I know that for $c=2$ it does. Also, it'd make it easier to generalize the above to other product series I have in mind. For instance, consider the possibility that the above product could be a specific example of a more general function.
P.S. I'd love to name it myself, if there isn't a name for it. Where would I go?
P.P.S Would I have better luck on MO?