In this paper, we define n th order α-difference operator and its equation and derive the corresponding discrete version of Leibnitz's and Montmorte's Theorems. Also, we investigate the numerical and complete solutions of n th order α-difference equation for finding the values of the sum of general partial sums of various finite series on arithmetic-geometric progression in the field of finite difference methods.
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.
In this paper, we introduce second order difference operator with logarithmic function and its inverse by which we obtain Fibonacci sequence and its sum. Some theorems and interesting results on the sum of the terms of advanced Fibonacci sequence are derived. Suitable examples are provided to illustrate our results and verified by using MATLAB.
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.
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