The generalized harmonic-oscillator algebra with
$aa^\dagger=\varphi(N)$ and $a^\dagger a=\varphi(N+1)$ giving
energies in closed form $E_n=(\varphi(n)+\varphi(n+1))/2$ is
reanalyzed as possibly giving energies via linear $(k+1)-$term
recurrences $E_{n+1}=\lambda_0E_n+\ldots + \lambda_{k-1}E_{n+1-k}$
with constant coefficients. For $\varphi(N)= N \times P(N)$ where
$P(N)=1+\mu_1N + \ldots + \mu_rN^r$ (and $r=k-2$) the authors
evaluate the ($\mu_j-$independent!) coefficients $\lambda_{j}$. It
is emphasized that the transition to inhomogeneous recurrences
re-introduces the $\mu_j-$dependence into the coefficients. The
alternative possibility of specifying energies via (for simplicity,
three-term) recurrences with non-constant coefficients is also
mentioned. A generalization of the construction is shown valid when
replacing $n$ by deformed $[n]_q$. In contrast, the further
transition from $[n]_q$ to $[n]_{p,q}$ (recall Eq. (38) for
definition) breaks the analogy. Two constructions (viz., with $r=1,
k=5$ and with $r=2, k=9$) lead to the conjecture that
$k=(r+2)(r+3)/2-1$ at $p\neq 1$.


MR2592328 Gavrilik, A. M.; Rebesh, A. P. Polynomially deformed
oscillators as $k$-bonacci oscillators. J. Phys. A 43 (2010), no. 9,
095203, 15 pp. 33D15 (81Q05)