Some Properties of Fibonacci Numbers, Generalized Fibonacci Numbers and Generalized Fibonacci Polynomial Sequences

Alexandre Laugier<br />Manjil P. Saikia

doi:10.5666/KMJ.2017.57.1.1

Article

Kyungpook Mathematical Journal 2017; 57(1): 1-84

Published online March 23, 2017

Some Properties of Fibonacci Numbers, Generalized Fibonacci Numbers and Generalized Fibonacci Polynomial Sequences

Alexandre Laugier¹
Manjil P. Saikia²

Lycée professionnel Tristan Corbiére, 16 rue de Kerveguen - BP 17149, 29671 Morlaix cedex, France¹
Department of Mathematical Sciences, Tezpur University, Napaam - 784028, Assam, India²Current Address: Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, Vienna 1090, Austria ²

Received: March 26, 2014; Accepted: May 13, 2015

Abstract

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Abstract

In this paper we study the Fibonacci numbers and derive some interesting properties and recurrence relations. We prove some charecterizations for F_p, where p is a prime of a certain type. We also define period of a Fibonacci sequence modulo an integer, m and derive certain interesting properties related to them. Afterwards, we derive some new properties of a class of generalized Fibonacci numbers. In the last part of the paper we introduce some generalized Fibonacci polynomial sequences and we derive some results related to them.

Keywords: Fibonacci numbers, congruences, period of Fibonacci sequence