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Polynomially deformed oscillators as k-bonacci oscillators

A family of multi-parameter, polynomially deformed oscillators (PDOs) given by polynomial structure function ϕ(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 n-th level energy E_n is given linearly, with real coefficients, by the two preceding ones E_{n-1}, E_{n-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 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 too: the (q;μ)-oscillator with ϕ(n) quadratic in the basic q-number [n]_q is shown to be Tribonacci one, while the (p,q;μ)-oscillators with ϕ(n) quadratic (cubic) in the p,q-number [n]_{p,q} are proven to obey the Pentanacci (Nine-bonacci) relations. Oscillators with general ϕ(n), polynomial in [n]_{q} or [n]_{p,q}, are also studied.