  Based on the q-deformed oscillator algebra, we study the behavior of the mean
occupation number and its analogies with intermediate statistics and we obtain
an expression in terms of an infinite continued fraction, thus clarifying
successive approximations. In this framework, we study the thermostatistics of
q-deformed bosons and fermions and show that thermodynamics can be built on the
formalism of q-calculus. The entire structure of thermodynamics is preserved if
ordinary derivatives are replaced by the use of an appropriate Jackson
derivative and q-integral. Moreover, we derive the most important thermodynamic
functions and we study the q-boson and q-fermion ideal gas in the thermodynamic
limit.
