Physical peculiarities of divergences emerging in q-deformed statistics

It was found in [Europhysics Letters {\bf 104}, (2013), 60003] that classical Tsallis theory exhibits poles in the partition function ${\cal Z}$ and the mean energy $<{\cal U}>$. These occur at a countably set of the q-line. We give here, via a simple procedure, a mathematical account of them. Further, by focusing attention upon the pole-physics, we encounter interesting effects. In particular, for the specific heat, we uncover hidden gravitational effects.