Composite bosons, here called quasibosons (e.g. mesons, excitons, etc.) occur in various physical situations. Quasibosons differ from bosons or fermions as their creation and annihilation operators obey non-standard commutation relations, even for the "fermion+fermion" composites. Our aim is to realize the operator algebra of quasibosons composed of two fermions or two q-fermions (q-deformed fermions) by the respective operators of deformed oscillators, the widely studied objects. For this, the restrictions on quasiboson creation/annihilation operators and on the deformed oscillator (deformed boson) algebra are obtained. Their resolving proves the uniqueness of the family of deformations and gives explicitly the deformation structure function (DSF) which provides the desired realization. In the case of two fermions as constituents, such realization is achieved when the DSF is quadratic polynomial in the number operator. In the case of two q-fermions, q = 1, the obtained DSF inherits the parameter q and does not continuously converge when q → 1 to the DSF of the first case.

We study the properties of sequences of the energy eigenvalues for some generalizations of q-deformed oscillators including the p,q-oscillator, the 3-, 4-and 5-parameter deformed oscillators given in the literature. It is shown that most of the considered models belong to the class of so-called Fibonacci oscillators for which any three consecutive energy levels satisfy the relation E n+1 = λE n +ρE n−1 with real constants λ, ρ. On the other hand, for certain µ-oscillator known from 1993 we prove the fact of its non-Fibonacci nature. Possible generalizations of the three-term Fibonacci relation are discussed among which we choose, as most adequate for the µ-oscillator, the so-called quasi-Fibonacci (or local Fibonacci) property of the energy levels. The property is encoded in the three-term quasiFibonacci (QF) relation with non-constant, n-dependent coefficients λ and ρ. Various aspects of the QF relation are elaborated for the µ-oscillator and some of its extensions.

A 3-parametric two-sided deformation of Heisenberg algebra (HA), with p, q-deformed commutator in the l.h.s. of basic defining relation and certain deformation of its r.h.s., is introduced and studied. The third deformation parameter µ appears in an extra term in the r.h.s. as pre-factor of Hamiltonian. For this deformation of HA we find novel properties. Namely, we prove it is possible to realize this (p, q, µ)-deformed HA by means of some deformed oscillator algebra. Also, we find the unusual property that the deforming factor µ in the considered deformed HA inevitably depends explicitly on particle number operator N . Such a novel N -dependence is special for the two-sided deformation of HA treated jointly with its deformed oscillator realizations.

We review on and give some further details about the thermodynamical properties of the µ-Bose gas model (arXiv:1309(arXiv: .1363) introduced by us recently. This model was elaborated in connection with µ-deformed oscillators. Here, we present the necessary concepts and tools from the so-called µ-calculus. For the high temperatures, we obtain the virial expansion of the equation of state, as well as five virial coefficients. In the regime of low temperatures, the critical temperature of condensation is inferred. We also obtain the specific heat, internal energy, and entropy for a µ-Bose gas for both low and high temperatures. All thermodynamical functions depend on the deformation parameter µ. The dependences of the entropy and the specific heat on the deformation parameter are visualized. K e y w o r d s: deformed oscillators, deformed analogs of the Bose gas model, µ-calculus, equation of state, virial coefficients, critical temperature, specific heat, entropy.

A new version of the q-deformed supersymmetric quantum mechanics (q-SQM), which is inspired by the Tamm-Dankoff-type (TD-type) deformation of quantum harmonic oscillator, is constructed. The obtained algebra of q-SQM is similar to that in Spiridonov's approach. However, within our version of q-SQM, the ground state found explicitly in the special case of superpotential yielding q-superoscillator turns out to be non-Gaussian and takes the form of special (TD-type) q-deformed Gaussian.