  Starting from the basic-exponential, a q-deformed version of the exponential
function established in the framework of the basic-hypergeometric series, we
present a possible formulation of a generalized statistical mechanics. In a
q-nonuniform lattice we introduce the basic-entropy related to the
basic-exponential by means of a q-variational principle. Remarkably, this
distribution exhibits a natural cut-off in the energy spectrum. This fact,
already encountered in other formulations of generalized statistical mechanics,
is expected to be relevant to the applications of the theory to those systems
governed by long-range interactions. By employing the q-calculus, it is shown
that the standard thermodynamic functional relationships are preserved,
mimicking, in this way, the mathematical structure of the ordinary
thermostatistics which is recovered in the q=1 limit.
