More generalized k(ε)-Fibonacci sequence, series, and its applications

In this study, we present a generalized higher-order delta operator with the co-efficient of falling factorial and its inverse, both of which allow us to get more generalized k(ε)-Fibonacci sequences along with their sums, a few theorems, and some intriguing conclusions regarding the sum of more generalized terms of k(ε)-Fibonacci sequences with falling factorial. In addition, the n-fold Fibonacci ratio and a few applications in life sciences and crystal growth were presented in the work. In addition, suitable examples are presented using MATLAB to show our results.