Generalized Oresme Numbers
Abstract
In this paper, we introduce the generalized Oresme sequence and we deal with, in detail, three special cases which we call them modified Oresme, Oresme-Lucas and Oresme sequences. We present Binet's formulas, generating functions, Simson formulas, and the summation formulas for these sequences. Moreover, we give some identities and matrices related with these sequences.
References
M. Clagett, 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.
M. Clagett, "Oresme, Nicole", in: C.C. Gillespie (ed.), Dictionary of Scientific Biography, Vol. 9, Charles Scribner's Sons, New York, 1981, pp. 223-230.
C.K. Cook, 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, pp. 87-99. https://doi.org/10.1007/978-0-306-48517-6_10
A.F. Horadam, Oresme numbers, Fibonacci Quarterly 12(3) (1974), 267-271.
A.F. Horadam, A generalized Fibonacci sequence, American Mathematical Monthly 68 (1961), 455-459. https://doi.org/10.1080/00029890.1961.11989696
A.F. Horadam, Basic properties of a certain generalized sequence of numbers, Fibonacci Quarterly 3.3 (1965), 161-176.
A.F. Horadam, Special properties of the sequence $w_{n}(a,b;p,q)$, Fibonacci Quarterly 5(5) (1967), 424-434.
A.F. Horadam, Generating functions for powers of a certain generalized sequence of numbers, Duke Math. J. 32 (1965), 437-446. https://doi.org/10.1215/S0012-7094-65-03244-8
M.C.S. Mangueira, R.P.M. Vieira, F.R.V. Alves and P.M.M.C. Catarino, The Oresme sequence: The generalization of its matrix form and its hybridization process, Notes on Number Theory and Discrete Mathematics 27(1) (2021), 101-111. https://doi.org/10.7546/nntdm.2021.27.1.101-111
N.J.A. Sloane, The on-line encyclopedia of integer sequences, http://oeis.org/
Y. Soykan, Simson identity of generalized m-step Fibonacci numbers, Int. J. Adv. Appl. Math. Mech. 7(2) (2019), 45-56.
Y. Soykan, On generalized (r,s)-numbers, Int. J. Adv. Appl. Math. Mech. 8(1) (2020), 1-14.
Y. Soykan, Some properties of generalized Fibonacci numbers: identities, recurrence properties and closed forms of the sum formulas $ sum_{k=0}^{n}x^{k}W_{mk+j}$, Archives of Current Research International 21(3) (2021), 11-38. https://doi.org/10.9734/acri/2021/v21i330235
This work is licensed under a Creative Commons Attribution 4.0 International License.
.jpg){kind=logo}
