On the $q-$parameter spectrum of generalized information-entropy measures with no cut-off prescriptions

After studying some properties of the generalized exponential and logarithmic function, in particular investigating the domain where the first maintains itself real and positive, and outlining how the known dualities $q \leftrightarrow \frac{1}{q}$ and $q \leftrightarrow 2-q$ play an important role, we shall examine the set of q-deforming parameters that allow generalized canonical maximum entropy probability distributions (MEPDs) to maintain itself positive and real without cut-off prescriptions. We determine the set of q-deforming parameters for which a generalized statistics with discrete but unbound energy states is possible.