Some Properties of the (p, q)‐Fibonacci and (p, q)‐Lucas Polynomials

GwangYeon Lee,Mustafa Asci

doi:10.1155/2012/264842

Abstract

Riordan arrays are useful for solving the combinatorial sums by the help of generating functions. Many theorems can be easily proved by Riordan arrays. In this paper we consider the Pascal matrix and define a new generalization of Fibonacci polynomials called (p, q)‐Fibonacci polynomials. We obtain combinatorial identities and by using Riordan method we get factorizations of Pascal matrix involving (p, q)‐Fibonacci polynomials.

Highlights

Large classes of polynomials can be defined by Fibonacci-like recurrence relation and yield Fibonacci numbers 1
The polynomials fn x studied by Catalan are defined by the recurrence relation fn x xfn−1 x fn−2 x, 1.1 where f1 x 1, f2 x x, and n ≥ 3
The Fibonacci polynomials studied by Jacobsthal were defined by

Summary

Research Article

Riordan arrays are useful for solving the combinatorial sums by the help of generating functions. Many theorems can be proved by Riordan arrays. In this paper we consider the Pascal matrix and define a new generalization of Fibonacci polynomials called p, q -Fibonacci polynomials. We obtain combinatorial identities and by using Riordan method we get factorizations of Pascal matrix involving p, q -Fibonacci polynomials

Introduction

By taking gF t parenthesis we get

Then we have n

By assumption we have

Now we can find the Riordan representation of the infinite matrix