Fibonacci Sequence and its Sum by Second Order Difference Operator with Polynomial Factorial

Sandra Pinelas,Rajiniganth P

doi:10.26524/cm36

Abstract

In this paper, we introduce second order difference operator with polynomial factorial and its inverse by which we obtain advanced Fibonacci sequence and its sum. Some theorems and interesting results on the sum of the terms of second order Fibonacci sequence are derived. Suitable examples are provided to illustrate our results and verified by MATLAB.

Highlights

We introduce second order difference operator with polynomial factorial ∆ v(k) = v(k) − α1k(p)v(k − 1) − α2k(q)v(k − 2), where α(k) = α1k(p), α1k(q) which α(k) generates α(k)-Fibonacci sequence and its sum
In 1984, Jerzy Popenda [5] introduced a particular type of difference operator on u(k) as ∆αu(k) = u(k + 1) − αu(k)
In 1989, Miller and Rose [8] introduced the discrete analogue of the Riemann-Liouville fractional derivative and its inverse ∆−h νf (t) ([1, 4])

Summary

Introduction

We introduce second order difference operator with polynomial factorial ∆ v(k) = v(k) − α1k(p)v(k − 1) − α2k(q)v(k − 2), where α(k) = α1k(p), α1k(q) which α(k) generates α(k)-Fibonacci sequence and its sum. We derive sum of the value of α(k)-Fibonacci Sequence by inverse of α(k) difference operator.

Results

Conclusion