Some properties of the generalized (p,q)- Fibonacci-Like number

Abstract

For the real world problems, we use some knowledge for explain or solving them. For example, some mathematicians study the basic concept of the generalized Fibonacci sequence and Lucas sequence which are the (p,q) – Fibonacci sequence and the (p,q) – Lucas sequence. Such as, Falcon and Plaza showed some results of the k-Fibonacci sequence. Then many researchers showed some results of the k-Fibonacci- Like number. Moreover, Suvarnamani and Tatong showed some results of the (p, q) - Fibonacci number. They found some properties of the (p,q) – Fibonacci number and the (p,q) – Lucas number. There are a lot of open problem about them. In this paper, we studied about the generalized (p,q)- Fibonacci-Like sequence. We establish properties like Catalan’s identity, Cassini’s identity, Simpson’s identity, d’Ocagne’s identity and Generating function for the generalized (p,q)-Fibonacci-Like number by using the Binet formulas. However, all results which be showed in this paper, are generalized of the (p,q) – Fibonacci-like number and the (p,q) – Fibonacci number. Corresponding author: kotmaster2@rmutt.ac.th