Iterated Binomial Transforms of the k-Fibonacci Sequence

Abstract

In this paper, we study the iterated application of the Binomial transform, the k–Binomial transform, the Rising k–Binomial transform, and the Falling k–Binomial transform to the k–Fibonacci sequence. In particular, we obtain the recurrence relation between the terms of the sequences obtained from these transforms and prove that they are all generalized Fibonacci sequences. As a consequence of this result, we obtain the Generating Function of these sequences, and Binnet Identity and Combinatorial Formula for the general term of each of them. We can consider the iterated application of the Binomial transforms as a new way to get integer sequences. But the way we have done, we have also found the recurrence relation between the terms of these sequences and how to find the general term of the same, either by the Binet Identity and the Combinatorial formula.