A study on dual hyperbolic generalized Pell numbers
Downloads
DOI:
https://doi.org/10.26637/mjm0903/005Abstract
In this paper, we introduce the generalized dual hyperbolic Pell numbers. As special cases, we deal with dual hyperbolic Pell and dual hyperbolic Pell-Lucas numbers. We present Binet's formulas, generating functions and the summation formulas for these numbers. Moreover, we give Catalan's, Cassini's, d'Ocagne's, Gelin-Ces\`{a}ro's, Melham's identities and present matrices related with these sequences.
Keywords:
Pell numbers, Pell-Lucas numbers, dual hyperbolic numbers, dual hyperbolic Pell numbers, Cassini identityMathematics Subject Classification:
06D50- Pages: 99-116
- Date Published: 01-07-2021
- Vol. 9 No. 03 (2021): Malaya Journal of Matematik (MJM)
M. AKAR, S. YÜCE AND S. ŞahIN, On the Dual Hyperbolic Numbers and the Complex Hyperbolic Numbers, Journal of Computer Science & Computational Mathematics, 8(1)(2018), 1-6.
J. BAEz, The octonions, Bull. Amer. Math. Soc. 39(2)(2002), 145-205.
N. BiCKNELl, A primer on the Pell sequence and related sequence, Fibonacci Quarterly, 13(4)(1975), 345349.
D.K. Biss, D. Dugger And D.C. IsAKsen, Large annihilators in Cayley-Dickson algebras, Communication in Algebra, 36(2)(2008), 632-664.
H.H.Cheng, And S. Thompson, Dual Polynomials and Complex Dual Numbers for Analysis of Spatial Mechanisms, Proc. of ASME 24th Biennial Mechanisms Conference, Irvine, CA, August, (1996), 19-22.
A. Cihan, A.Z. AZak, M.A GÜngör And M.Tosun, A Study on Dual Hyperbolic Fibonacci and FibonacciLucas Numbers, An. Şt. Univ. Ovidius Constanta, 27(1)(2019), 35-48.
J. Cockle, On a New Imaginary in Algebra, Philosophical magazine, London-Dublin-Edinburgh, 3(34)(1849), 37-47.
A. Dasdemir, On the Pell, Pell-Lucas and Modified Pell Numbers By Matrix Method, Applied Mathematical Sciences, 5(64)(2011), 3173-3181.
J. Ercolano, Matrix generator of Pell sequence, Fibonacci Quarterly, 17(1)(1979), 71-77.
P.Fjelstad and G. Gal Sorin, n-dimensional Hyperbolic Complex Numbers, Advances in Applied Clifford Algebras, 8(1)(1998), 47-68.
H. GÖKBAŞ AND H. KÖsE, Some sum formulas for products of Pell and Pell-Lucasnumbers, Int. J. Adv. Appl. Math. and Mech. 4(4)(2017), 1-4.
W.R. Hamilton, Elements of Quaternions, Chelsea Publishing Company, New York, (1969).
A.F. Horadam, Pell identities, Fibonacci Quarterly, 9(3)(1971), 245-263.
K. Imaeda and M. Imaeda, Sedenions: algebra and analysis, Applied Mathematics and Computation, $mathbf{1 1 5}(2000)$, 77-88.
I. Kantor And A.SolodovniKov, Hypercomplex Numbers, Springer-Verlag, New York, 1989.
E. Kiliç And D. TAşci, The Linear Algebra of The Pell Matrix, Boletín de la Sociedad Matemática Mexicana, 11(2)(2005), 163-174.
E. Kiliç And D. TAŞCI , The Generalized Binet Formula, Representation and Sums of the Generalized Order-k Pell Numbers, Taiwanese Journal of Mathematics, 10(6)(2006), 1661-1670.
E. Kiliç And P. StaniCa, A Matrix Approach for General Higher Order Linear Recurrences, Bulletin of the Malaysian Mathematical Sciences Society 34(1)(2011), 51-67.
T. Koshy, Pell and Pell-Lucas Numbers with Applications, Springer, New York, (2014).
R. Melham, Sums Involving Fibonacci and Pell Numbers, Portugaliae Mathematica, 56(3)(1999), 309317.
G. Moreno, The zero divisors of the Cayley-Dickson algebras over the real numbers, Bol. Soc. Mat. Mexicana 3(4)(1998), 13-28.
N.J.A. SlOANE, The on-line encyclopedia of integer sequences, http://oeis.org/.
G. Sовсzук, The Hyperbolic Number Plane, The College Mathematics Journal, 26(4)(1995), 268-280.
Y. Soykan, On Generalized Third-Order Pell Numbers, Asian Journal of Advanced Research and Reports, 6(1)(2019), 1-18.
Y. Soykan, A Study of Generalized Fourth-Order Pell Sequences, Journal of Scientific Research and Reports, 25(1-2)(2019), 1-18.
Y. Soykan, Tribonacci and Tribonacci-Pell-Lucas Sedenions. Mathematics 7(1)(2019), 74.
Y. Soykan, On Summing Formulas For Generalized Fibonacci and Gaussian Generalized Fibonacci Numbers, Advances in Research, 20(2)(2019), 1-15.
Y. Soykan, On Dual Hyperbolic Generalized Fibonacci Numbers, Preprints (2019), 2019100172 (doi: 10.20944/preprints201910.0172.v1).
T. YAĞMUR, New Approach to Pell and Pell-Lucas Sequences, Kyungpook Math. J., 59(2019), 23-34.
- NA
Metrics
Published
How to Cite
Issue
Section
License
Copyright (c) 2021 Yüksel Soykan, Mehmet Gümüş, Melih Göcen
This work is licensed under a Creative Commons Attribution 4.0 International License.
