  We study analytical aspects of a generic q-deformation with q real, by
relating it with discrete scale invariance. We show how models of conformal
quantum mechanics, in the strong coupling regime and after regularization, are
also discrete scale invariant. We discuss the consequences of their distinctive
spectra, characterized by functional behavior. The role of log-periodic
behavior and q-periodic functions is examined, and we show how q-deformed zeta
functions, characterized by complex poles, appear. As an application, we
discuss one-loop effects in discretely self-similar space-times.
