On dimensions of oscillator algebras

On the dimensions of oscillator-like algebras induced by orthogonal polynomials: Nonsymmetric case

“…In [7] the authors have presented some remarks about the dimensions of oscillator algebras associated with two dimensional orthogonal polynomials such as the normalized 2D-Hermite polynomials H n,m (z, z) which satisfy the three term recurrence relation [13,17] (5.1) zH m,n (z, z) = √ m + 1H m+1,n (z, z) + √ nH m,n−1 (z, z).…”

“…Remark 2.5. At this point we like to respond to the comments made in [7] and [8] about our earlier paper [14]. In [7] and [8] the authors claimed that the sufficient part of the Theorem (2.4) is incorrect.…”

“…At this point we like to respond to the comments made in [7] and [8] about our earlier paper [14]. In [7] and [8] the authors claimed that the sufficient part of the Theorem (2.4) is incorrect. They indicated that for the sufficient part to be true, in addition to (2.11), the coefficients α 0 , α 1 and α 2 of (2.11) must also satisfy the relation…”

Orthogonal polynomials and deformed oscillators

“…In [29] considered only case of orthogonal polynomials for a symmetric measure on the real axis (when the Jacobi matrix corresponding to the recurrent relations (1) has zero diagonal). In the work [30] we have clarified the formulation of sufficient conditions and extended the results to the case of the Jacobian matrix with nonzero diagonal, corresponding to recurrent relations (2). Following property holds [30]:…”

“…In the work [30] we have clarified the formulation of sufficient conditions and extended the results to the case of the Jacobian matrix with nonzero diagonal, corresponding to recurrent relations (2). Following property holds [30]:…”