Horadam Numbers: Sum Formulas \(\sum_{k=0}^{n} k x^{k} W_{k}\) and \(\sum_{k=1}^{n} k x^{k} W_{-k}\)
Yuksel Soykan *
Department of Mathematics, Art and Science Faculty, Zonguldak B¨ ulent Ecevit University, 67100, Zonguldak, Turkey.
*Author to whom correspondence should be addressed.
Abstract
In this paper, closed forms of the sum formulas \(\sum_{k=0}^{n} k x^{k} W_{k}\) and \(\sum_{k=1}^{n} k x^{k} W_{-k}\) for generalized Fibonacci numbers are presented. As special cases, we give sum formulas of Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas numbers. We present the proofs to indicate how these formulas, in general, were discovered. Of course, all the listed formulas may be proved by induction, but that method of proof gives no clue about their discovery.
Keywords: Isoetes sporogenesis, Fibonacci numbers, Irregular meiosis, Lucas numbers, Pell numbers, Jacobsthal numbers, sum formulas