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ON ao-GENERALIZED FIBONACCI SEQUENCES
| Content Provider | Semantic Scholar |
|---|---|
| Author | Motta, Walter Rachidi, M. Saeki, Osamu |
| Copyright Year | 2010 |
| Abstract | When at = 1 for all / and A = (0,0, ...,0,1), we get the r-generalized Fibonacci numbers (see [4]). A Binet-type formula and a combinatorial expression of weighted r-generalized Fibonacci sequences are given in [3]. Furthermore, in [2], the convergence of the sequence { jVU^/^V 2 } has been studied, where q is a root of the characteristic polynomial P(x) -xa0x~ -ar_2x-ar_l of multiplicity v. The purpose of this paper is to generalize the weighted r-generalized Fibonacci sequences with 1 < r < oo to a class of sequences which are defined by recurrence formulas involving infinitely many terms, and to analyze their asymptotic behavior. We call such sequences ^-generalized Fibonacci sequences. This is a new generalization of the usual Fibonacci sequences and almost nothing has been known about such sequences until now. For example, there has been no theory of difference equations for such sequences. More precisely, an oo-generalized Fibonacci sequence is defined as follows. We suppose that two infinite sequences of complex numbers are given, one for the initial sequence and the other for the weight sequence. Then a member of the oo-generalized Fibonacci sequence is determined by the weighted series of its preceding members (for a precise definition, see §2). Since the recurrence formula always involves infinitely many terms, we always have to worry about the convergence of the series corresponding to the recurrence formula and hence we need auxiliary conditions on the initial sequence and the weight sequence. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://www.fq.math.ca/Scanned/37-3/motta.pdf |
| Alternate Webpage(s) | http://www.mathstat.dal.ca/FQ/Scanned/37-3/motta.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
