Some Boundedness, Convergence, and Oscillatory Results of Generalized Fibonacci-Type and Fibonacci Ratio Recurrences

Some boundedness and convergence properties of generalized Fibonacci’s-type recurrences and their associated iterated recurrence ratios between pairs of consecutive terms are discussed under a wide number of initial conditions. Also, a more general, so-called k,q Fibonacci’s recurrence and the associated Fibonacci’s ratio recurrences are investigated, where the constants k and q are the prefixed gains used to generate each new member of the recurrences from the two preceding ones. Some oscillation and periodicity conditions are also discussed depending on the initial conditions, while convergence and stability properties are also dealt with. The initial conditions of the recurrences can be fixed arbitrarily, so that both the well-known standard Fibonacci recurrence and the Lucas recurrence are just particular cases. In the most general case, the initial conditions can be integer or real and of alternate signs. Later on, the manuscript deals with some further boundedness, oscillation, convergence, and stability properties of generalized Fibonacci recurrences and associated Fibonacci ratio recurrences being generated under, in general, recurrence-dependent gain sequences.