On the Dual δ − k − Fibonacci Numbers

Abstract

We define two integer sequences that depend on a parameter and that are related to each other by two recurrence relations. Then we find the Binet formula for the terms of these sequences and, by developing it, we will get an equivalent combinatorial formula. We show that each sequence follows the same relation of recurrence although they differ in the initial conditions. Later we show that these numbers are related to the k − Fibonacci numbers and we finish this section nding its generating functions. Finally, for certain particular values of δ we show that these numbers are related to the Chebyshev polynomials. This paper deals with a new concept of k − Fibonacci sequences linked to each other, so there is no literature on the subject. I hope that this article will be the starting point for other mathematicians that wish to investigate this topic.