Polynomially deformed oscillators as k-bonacci oscillators

A family of multi-parameter, polynomially deformed oscillators (PDOs) given by the polynomial structure function ψ(n) is studied from the viewpoint of being (or not) in the class of Fibonacci oscillators. These obey the Fibonacci relation/property (FR/FP) meaning that the nth level energy En is given linearly, with real coefficients, by the two preceding ones En-1, En-2. We first prove that the PDOs do not fall in the Fibonacci class. Then, three different paths of generalizing the usual FP are developed for these oscillators: we prove that the PDOs satisfy the respective k-term generalized Fibonacci (or 'k-bonacci') relations; for these same oscillators we examine two other generalizations of the FR, the inhomogeneous FR and the 'quasi-Fibonacci' relation. Extended families of deformed oscillators are studied as well: the (q; μ)-oscillator with ψ(n) quadratic in the basic q-number [n]q is shown to obey the Tribonacci relation, while the (p, q; μ)-oscillators with ψ(n) quadratic (cubic) in the p, q-number [n]p,q are proven to obey the Pentanacci (Nine-bonacci) relations. Oscillators with general ψ(n), polynomial in [n]q or [n]p,q, are also studied.