Study About Half Self–convolution of the k–Fibonacci Sequence

We say the k-Fibonacci numbers Fk,i and Fk,j are equidistant if j = n - i and then we study some properties of these pairs of numbers. As a main result, we look for the formula to find the generating function of the product of the equidistant numbers, their sums and their binomial transforms. Next, we apply this formula to some simple cases but more common than the general cases. In particular, we define the half-self-convolution of the k-Fibonacci and k-Lucas sequences. Finally, we study the sum of these new sequences, their recurrence relations, and their generating functions.