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

TL;DR

This paper explores properties, recurrence relations, and characterizations of Fibonacci numbers, generalized Fibonacci numbers, and Fibonacci polynomial sequences, including modular periods and new properties of these sequences.

Contribution

It introduces new properties and characterizations of Fibonacci and generalized Fibonacci sequences, including polynomial generalizations and modular period analysis.

Findings

Derived new recurrence relations for Fibonacci numbers

Characterized Fibonacci numbers at prime indices

Analyzed periods of Fibonacci sequences modulo integers

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.