We show that a natural realization of the thermostatistics of q bosons can be built on the formalism of q calculus, and that the entire structure of thermodynamics is preserved if we use an appropriate Jackson derivative in place of the ordinary thermodynamics derivative. This framework allows us to obtain a generalized q boson entropy which depends on the q basic number. We study the ideal q boson gas in the thermodynamic limit which is shown to exhibit Bose-Einstein condensation with a higher critical temperature and a discontinuous specific heat.

We study the thermostatistics of q-deformed bosons and fermions obeying the symmetric (q<-->q(-1)) algebra and show that it can be built on the formalism of q calculus. The entire structure of thermodynamics is preserved if ordinary derivatives are replaced by an appropriate Jackson derivative. In this framework, we derive the most important thermodynamic functions describing the q-boson and q-fermion ideal gases in the thermodynamic limit. We also investigate the semiclassical limit and the low-temperature regime and demonstrate that the nature of the q deformation gives rise to pure quantum statistical effects stronger than undeformed boson and fermion particles.

In the framework of the q-deformed oscillator algebra we investigate the nature of the interaction among the particles of a free q-boson gas with a view to interpreting the interaction in terms of the general properties of the qdeformation. It is shown that many thermodynamic quantities characterizing the q-boson system can be expressed as power series in the number of particles operator N or as a series in the deformation parameter q, implying that the q-deformation can originate from the interaction among the particles of the ensemble. As an example, we derive the virial expansion of the q-boson gas, showing that the lowest order coefficient reduces to the standard virial coefficient of an undeformed boson gas, while the higher order coefficients contain effects arising from interaction.

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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 qnonuniform 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.

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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 qcalculus. 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.