On the Sum of the Cubes of Generalized Oresme Numbers: The Sum Formula \(\sum_{k=0}^{n} x^{k} W_{m k+j}^{3}\)
Yüksel 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 \(\sum_{k=0}^{n} x^{k} W_{m k+j}^{3}\) for generalized Oresme numbers are presented. As special cases, we give sum formulas of modified Oresme, Oresme-Lucas and Oresme 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: Leaf extracts, Modified Oresme numbers, effect, Effect of Parthenium, Oresme-Lucas numbers, Stomatal features, Oresme numbers, sum formulas