Abstract
In this paper, we first obtain a series of k-Lucas numbers using k-Lucas numbers. We give new presentations of any real number u ≠ 0 using this obtained k-Lucas series and show polynominal repsesentations that every nonzero real number can be uniquely represented as the sum of the squares of consecutive k-Lucas numbers. To do this, we give new presentation theorems for any real number using k-Lucas series. Finally, to support these theorems, we give examples where we obtain the roots of the polynomial representations of a selected real number u ≠ 0, as well as the values representing the first ten prime numbers corresponding to a chosen k-Lucas polynomial.
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