On Periodic ∞-generalized Fibonacci Sequences

The notion of an∞-generalized Fibonacci sequence has been introduced in [6], and studied in [1], [7], [9]. This class of sequences defined by linear recurrences of infinite order is an extension of the class of ordinary (weighted) r-generalized Fibonacci sequences (r-GFS, for short) with r finite defined by linear recurrences of r order (for example, see [2], [3], [4], [5], [8] etc.) More precisely, let {ai}i=0 and {α−i}i=0 be two sequences of real or complex numbers, where ai 6= 0 for some i. The former is called the coefficient sequence and the latter the initial sequence. The associated ∞-generalized Fibonacci sequence (∞-GFS, for short) {Vn}n∈Z is defined as follows: