International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 50, Pages 2681-2693

Extended Fibonacci numbers and polynomials with probability applications

Demetrios L. Antzoulakos

Department of Statistics and Insurance Science, University of Piraeus, 80 Karaoli & Dimitriou street, Piraeus 18534, Greece

Received 30 April 2004

Copyright © 2004 Demetrios L. Antzoulakos. 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.


The extended Fibonacci sequence of numbers and polynomials is introduced and studied. The generating function, recurrence relations, an expansion in terms of multinomial coefficients, and several properties of the extended Fibonacci numbers and polynomials are obtained. Interesting relations between them and probability problems which take into account lengths of success and failure runs are also established.