Abstract
The hybrid numbers were introduced by Ozdemir [9] as a new generalization of complex, dual, and hyperbolic numbers. A hybrid number is defined by \(k=a+bi+c\epsilon +dh\), where a, b, c, d are real numbers and \( i,\epsilon ,h\) are operators such that \(i^{2}=-1,\epsilon ^{2}=0,h^{2}=1\) and \(ih=-hi=\epsilon +i\). This work is intended as an attempt to introduce the bi-periodic Horadam hybrid numbers which generalize the classical Horadam hybrid numbers. We give the generating function, the Binet formula, and some basic properties of these new hybrid numbers. Also, we investigate some relationships between generalized bi-periodic Fibonacci hybrid numbers and generalized bi-periodic Lucas hybrid numbers.
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References
Bilgici G. (2014). Two generalizations of Lucas sequence. Appl. Math. Comput., 245, 526-538.
Catarino P. (2019). On \(k\)-Pell hybrid numbers. J Discrete Math Sci Cryptography, 22(1), 83-89.
Edson M., Yayenie O. (2009). A new generalizations of Fibonacci sequences and extended Binet’s formula, Integers 9, 639-654.
Halici S., Karatas A. (2017). On a generalization for quaternion sequences, Chaos Solitons Fractals, 98, 178-182.
Hamilton W.R. (1853). Lectures on quaternions. Hodges and Smith. Dublin.
Horadam A.F. (1963). Complex Fibonacci numbers and Fibonacci quaternions, Amer. Math. Montly, 70, 289-291.
Horadam A.F. (1965). Basic properties of a certain generalized sequence of numbers, The Fibonacci Quarterly, 3(3) 161-176.
Morales G.C. (2021). Investigation of generalized hybrid Fibonacci numbers and their properties. Applied Mathematics E-Notes , 21, 110-118.
Ozdemir M. (2018). Introduction to hybrid numbers, Advances in Applied Clifford Algebras, 28(1), 11.
Sahin M. (2011). The Gelin-Cesaro identity in some conditional sequences, Hacet. J. Math. Stat., 40 (6), 855-861.
Senturk T.D., Bilgici G., Dasdemir A. and Unal Z. (2020). A study on Horadam hybrid numbers, Turk J Math., 44, 1212-1221.
Szynal-Liana A. (2018). The Horadam hybrid numbers, Discuss Math Gen Algebra Appl. 38 (1), 91-98.
Szynal-Liana A., Wloch I. (2019). The Fibonacci hybrid numbers, Utilitas Math., 110, 3-10.
Szynal-Liana A., Wloch I. (2019). On Jacobsthal and Jacobsthal-Lucas hybrid numbers, Ann Math Silesianae, 33(1), 276-283.
Szynal-Liana A., Wloch I. (2018). On Pell and Pell-Lucas hybrid numbers, Commentat Math. 58, 11-17.
Tan E., Leung H-H. (2020). A note on congruence properties of the generalized bi-periodic Horadam sequence, Hacet. J. Math. Stat. 49 (6), 2084-2093.
Tan E., Leung H-H. (2020). Some basic properties of the generalized bi-periodic Fibonacci and Lucas sequences, Advances in Difference Equations, 26.
Yayenie O. (2011). A note on generalized Fibonacci sequence, Appl. Math. Comput., 217, 5603-5611.
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Communicated by B. Sury.
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Tan, E., Ait-Amrane, N.R. On a new generalization of Fibonacci hybrid numbers. Indian J Pure Appl Math 54, 428–438 (2023). https://doi.org/10.1007/s13226-022-00264-3
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DOI: https://doi.org/10.1007/s13226-022-00264-3