A Study on the Sum of the Squares of Generalized Oresme Numbers: The Sum Formula \(\sum_{k=0}^{n} x^{k} W_{m k+j}^{2}\)
Yuksel Soykan *
Department of Mathematics, Art and Science Faculty, Zonguldak Bulent Ecevit University, 67100, Zonguldak, Turkey.
*Author to whom correspondence should be addressed.
Abstract
In this study, closed forms of the sum formulas \(\sum_{k=0}^{n} x^{k} W_{m k+j}^{2}\) for generalized Oresme numbers are given. As special cases, we present sum formulas of Modified Oresme, Oresme-Lucas and Oresme numbers. We present the proofs to indicate how these formulas, in general, were discovered. All the listed formulas may be proved by induction, but that method of proof gives no clue about their discovery.
Keywords: Laser, Modified Oresme numbers, biostimulation, Oresme-Lucas numbers, photoreceptors grains, Oresme numbers, agriculture, sum formulas