Physics – Mathematical Physics
Scientific paper
2009-10-14
J. Phys. A: Math. Theor. 43 (2010) 095203 (15pp)
Physics
Mathematical Physics
16 pages; minor revisions, references added, to appear in J. Phys. A: Math. Theor
Scientific paper
10.1088/1751-8113/43/9/095203
A family of multi-parameter, polynomially deformed oscillators (PDOs) given by polynomial structure function \phi(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;\mu)-oscillator with \phi(n) quadratic in the basic q-number [n]_q is shown to be Tribonacci one, while the (p,q;\mu)-oscillators with \phi(n) quadratic (cubic) in the p,q-number [n]_{p,q} are proven to obey the Pentanacci (Nine-bonacci) relations. Oscillators with general \phi(n), polynomial in [n]_{q} or [n]_{p,q}, are also studied.
Gavrilik Alexandre M.
Rebesh Anastasiya P.
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