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
How to Cite
Downloads
References
Clagett M. Nicole Oresme and the Medieval Geometry of Qualities and Motions: A Treatise on the Uniformity and Difformity of Intensities Known as Tractatus de confurationibus qualitatum et motuum, The University of Wisconsin Press, Wisconsin; 1968.
Clagett M. ”Oresme, Nicole”, C.C. Gillespie (ed.), Dictionary of Scientific Biography, Charles Scribner’s Sons, New York. 1981;9:223-230. Cook CK. Some sums related to sums of Oresme numbers, In: Howard F. T. (eds) Applications of Fibonacci Numbers, volume 9, Proceedings of the Tenth International Research Conference on Fibonacci Numbers and their Applications, Kluwer Academic Publishers. 2004;87-99.
Horadam AF. Oresme Numbers. Fibonacci Quarterly. 1974;12(3):267-271.
Mangueira MCS, Vieira RPM, Alves FRV, Catarino PMC. The Oresme sequence: The generalization of its matrix form and its hybridization process. Notes on Number Theory and Discrete Mathematics. 2021;27(1):101-111.
DOI: 10.7546/nntdm.2021.27.1.101-111.
Soykan Y. Generalized Oresme Numbers. Earthline Journal of Mathematical Sciences. 2021;7(2):333-367.
Available:https://doi.org/10.34198/ejms.7221.333367
Soykan Y. Generalized Fibonacci Numbers with Indices in Arithmetic Progression and Sum of Their Squares: The Sum Formula Σn k=0 xkW2 mk+j ; Journal of Advances in Mathematics and Computer Science. 2021;36(6):30-62.
ISSN: 2456-9968. DOI: 10.9734/JAMCS/2021/v36i630371