With this concept closely linked:

1\) The Hilbert space $\mathcal{H}$ of oscillator states; the classical Hermite polynomials, which are orthogonal with respect to the Gaussian measure on the real axis and forms the basis in the space $\mathcal{H}$; the fundamental quantum-mechanical operators in $\mathcal{H}$ — the operators of the coordinate, momentum, and quadratic Hamiltonian.

2)The Fock space and main operators in this space — ladder operators, the operator of the number of particles, and identical operator, which are the generators of the oscillator algebra (algebra of dynamical symmetries for the quantum system).

Development of quantum physics at the end of the last century, in particular, the emergence of quantum algebras [@N01]-[@N04], has resulted in various generalizations of the quantum harmonic oscillator. The first meaningful generalization was the $q$-oscillator [@N05]-[@N09] connected with $q$-deformation of canonical commutation relations of the harmonic-oscillator algebra. The next step in generalization the quantum-mechanical oscillator was the construction of a ladder operators that satisfy certain commutation relations (i.e. construction of an oscillator-like algebra that generalizes the Heisenberg algebra [@N10]-[@N12]) connected with some polynomials from the Askey - Wilson scheme [@N13]-[@N15]. Analysis of the oscillator algebras related to some known (mostly classical) orthogonal polynomials allowed us to develop 