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

Abstract

In this paper we study the Fibonacci numbers and derive some interesting properties and recurrence relations. We prove some charecterizations for Fp, 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.

Some new and old Gibonacci identities

“…For example, we have G n ≡ G n+3 (mod 2), G n ≡ G n+8 (mod 3), G n ≡ G n+20 (mod 5), G n ≡ G n+16 (mod 7), G n ≡ G n+10 (mod 11), G n ≡ G n+14 (mod 29), etc. Results of these type for Fibonacci and generalized Fibonacci type sequences were derived using noncombinatorial techniques by Laugier and the second author [LS17]. (5) Another aspect which is immediately clear is that we can extend some of our proof techniques to three or more term recurrences.…”