Fourier transform (similar to the transformation in the work [@N16]), based on the use of the Poisson kernel. Next, note that the correct choice of “coordinate” operator $X$ for radial oscillator in Sec.2.2.2 (see ) is not as easy as for sectorial oscillator. The mentioned choice demands prior to obtain the recurrence relations for the ChK-polynomials in the “radial direction”. The momentum operator $X^\dag$ is constructed using the corresponding Poisson kernel . This results in the non-standard form for the ladder operators and Hamiltonian of the radial oscillator. The additional term in can be considered as “interaction energy” of the sectorial and radial oscillators.

Secondly, the introduction in the Subsection 2.2.3 one more “marginal” oscillator due to the fact that according to the the union $\mathfrak{A}_{s,r}$ of the sectorial and radial oscillator algebras decomposes into the direct sum $\mathfrak{A}_{s,r}=\mathfrak{A}^{even}_{s,r}\oplus\mathfrak{A}^{odd}_{s,r}$. Thus it is clear that $\mathfrak{A}_{s,r}$ is a subalgebra of the desired algebra of two-dimensional ChK-oscillator. So that we must introduce additional oscillator in Subsection 2.2.3, which we call the “boundary” oscillator, because the generators of the corresponding oscillator algebra are not trivial only in “boundary subspaces” $\Hco$ and $\Hoc$ (see ).

Further, we note that all constructed in section 2.2 