Advanced fibonacci sequence and its sum by second order variable co-efficient difference operator

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 (α₁t^r1, α₂t^r2 )-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 are
presented.