Advances in Difference Equations
Volume 2006 (2006), Article ID 23849, 11 pages
doi:10.1155/ADE/2006/23849

Extending generalized Fibonacci sequences and their binet-type formula

Mustapha Rachidi1 and Osamu Saeki2

1Section de Mathématique, LEGT - F. Arago, Académie de Reims, 1, rue F. Arago, Reims 51100, France
2Faculty of Mathematics, Kyushu University, Hakozaki 812-8581, Fukuoka, Japan

Received 8 May 2006; Accepted 2 July 2006

Copyright © 2006 Mustapha Rachidi and Osamu Saeki. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the extension problem of a given sequence defined by a finite order recurrence to a sequence defined by an infinite order recurrence with periodic coefficient sequence. We also study infinite order recurrence relations in a strong sense and give a complete answer to the extension problem. We also obtain a Binet-type formula, answering several open questions about these sequences and their characteristic power series.