Abstract
We introduce a second order difference operator with specific powers of variable co-efficient and its inverse in this study, which allows us to derive the (α1tr1, α2tr2 )-Fibonacci sequence and its summation. This series is known as the Fibonacci sequence with variable co-efficients (VCFS). On the sum of the terms of the variable co-efficient Fibonacci sequence, some theorems and intriguing findings are generated. To demonstrate our findings, appropriate instances arepresented.
Highlights
IntroductionG.Britto Antony Xavier, et al, [3] presented the second order α−difference operator as ∆α( ,m)v(t) = v(k + 2 ) − α[v(k + ) + v(k + m)] − α2v(t) in 2014, and discovered a finite series solution to the associated generalised second order difference equation ∆α( ,m)v(t) = u(t)
Jerzy Popenda [6] created a new form of difference operator on u(t) in 1984
By introducing the (α1tr1, α2tr2)-difference operator, we were able to obtain the summation formula for the (α1tr1, α2tr2)-Fibonacci sequence, and we were able to derive certain results on the closed and summation form solution of the second order difference equation, which will be used in our future research
Summary
G.Britto Antony Xavier, et al, [3] presented the second order α−difference operator as ∆α( ,m)v(t) = v(k + 2 ) − α[v(k + ) + v(k + m)] − α2v(t) in 2014, and discovered a finite series solution to the associated generalised second order difference equation ∆α( ,m)v(t) = u(t). With this background, we used the difference Operator with variable co-efficients to produce advanced Fibonacci sequence and its sum in this study
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