Some phase-transitions and their q-analogs on powers of the quantum dilogarithm

The quantum dilogarithm has been important object to study in mathematics and physics. We consider phase-transitions on the powers of the quantum dilogarithm, assuming statistical-mechanical interpretations on all of these powers. We discuss the phase-transitions by its multiplicative q-analog. We then state some conjectural positivity of q-polynomials and additive q-analogs of the phase-transitions. This is a work-in-progress.