On Dual-Complex Numbers with Generalized Fibonacci and Lucas Numbers Coefficients

Arzu Cihan; Ayşe Zeynep Azak; Mehmet Ali Güngör

doi:10.36753/mathenot.621602

EN

On Dual-Complex Numbers with Generalized Fibonacci and Lucas Numbers Coefficients

Abstract

In this paper, dual-complex Fibonacci numbers with generalized Fibonacci and Lucas coefficients are dened. Generating function is given for this number system. Binet formula is obtained by the help of this generating function. Then, well-known Cassini, Catalan, d'Ocagne's, Honsberger, Tagiuri and other identities are given for this number system. Finally, it is seen that the theorems and the equations which are obtained for the special values p = 1 and q = 0 correspond to the theorems and identities in [2].

Keywords

Dual-complex numbers,generalized Fibonacci

References

[1] Dunlap R.A.: The Golden Ratio and Fibonacci Numbers. World Scientific Publishing Co. Pte. Ltd., Singapore (1997).
[2] Gungor M.A. and Azak A.Z.: Investigation of Dual-complex Fibonacci, Dual-complex Lucas Numbers and Their Properties. Adv. Appl. Clifford Algebras 27, 3083{3096, (2017).
[3] Hoggatt Jr. V.E.: Fibonacci and Lucas Numbers. Houghton-Mifflin Co.,Boston (1969).
[4] Horadam A.F.: A Generalized Fibonacci Sequence. Amer. Math. Monthly 68, 455{459, (1961).
[5] Iakini A.L.: Generalized Quaternions of Higher Order.Fibonacci Quart. 15, 343{346, (1977).
[6] Koshy T.: Fibonacci and Lucas Numbers with Applications. Wiley and Sons Publication, New York (2001).
[7] Majernik V.: Multicomponent Number Systems. Acta Phys. Pol. A 90(3), 491{498 (1996).
[8] Messelmi F.: Dual-complex Numbers and Their Holomorphic Functions. https://hal.archives-ouvertes.fr/hal-01114178, (2015).
[9] Scholfield P.H.: The Theory of Proportion in Architecture. Cambridge University Press, Cambridge (1958).
[10] Silvester J.R.: Fibonacci Properties by Matrix Methods.Math. Gaz. 63(425), 188{191, (1979).

Details

Primary Language

English

Subjects

Engineering

Journal Section

Research Article

Authors

Arzu Cihan This is me
Türkiye

Ayşe Zeynep Azak
Türkiye

Mehmet Ali Güngör ^*
0000-0003-1863-3183
Türkiye

Publication Date

March 20, 2020

Submission Date

September 18, 2019

Acceptance Date

February 14, 2020

Published in Issue

Year 2020 Volume: 8 Number: 1

DOI

https://doi.org/10.36753/mathenot.621602

IZ

https://izlik.org/JA35RJ53FR

APA

Cihan, A., Azak, A. Z., & Güngör, M. A. (2020). On Dual-Complex Numbers with Generalized Fibonacci and Lucas Numbers Coefficients. Mathematical Sciences and Applications E-Notes, 8(1), 55-68. https://doi.org/10.36753/mathenot.621602

AMA

1.Cihan A, Azak AZ, Güngör MA. On Dual-Complex Numbers with Generalized Fibonacci and Lucas Numbers Coefficients. Math. Sci. Appl. E-Notes. 2020;8(1):55-68. doi:10.36753/mathenot.621602

Chicago

Cihan, Arzu, Ayşe Zeynep Azak, and Mehmet Ali Güngör. 2020. “On Dual-Complex Numbers With Generalized Fibonacci and Lucas Numbers Coefficients”. Mathematical Sciences and Applications E-Notes 8 (1): 55-68. https://doi.org/10.36753/mathenot.621602.

EndNote

Cihan A, Azak AZ, Güngör MA (March 1, 2020) On Dual-Complex Numbers with Generalized Fibonacci and Lucas Numbers Coefficients. Mathematical Sciences and Applications E-Notes 8 1 55–68.

IEEE

[1]A. Cihan, A. Z. Azak, and M. A. Güngör, “On Dual-Complex Numbers with Generalized Fibonacci and Lucas Numbers Coefficients”, Math. Sci. Appl. E-Notes, vol. 8, no. 1, pp. 55–68, Mar. 2020, doi: 10.36753/mathenot.621602.

ISNAD

Cihan, Arzu - Azak, Ayşe Zeynep - Güngör, Mehmet Ali. “On Dual-Complex Numbers With Generalized Fibonacci and Lucas Numbers Coefficients”. Mathematical Sciences and Applications E-Notes 8/1 (March 1, 2020): 55-68. https://doi.org/10.36753/mathenot.621602.

JAMA

1.Cihan A, Azak AZ, Güngör MA. On Dual-Complex Numbers with Generalized Fibonacci and Lucas Numbers Coefficients. Math. Sci. Appl. E-Notes. 2020;8:55–68.

MLA

Cihan, Arzu, et al. “On Dual-Complex Numbers With Generalized Fibonacci and Lucas Numbers Coefficients”. Mathematical Sciences and Applications E-Notes, vol. 8, no. 1, Mar. 2020, pp. 55-68, doi:10.36753/mathenot.621602.

Vancouver

1.Arzu Cihan, Ayşe Zeynep Azak, Mehmet Ali Güngör. On Dual-Complex Numbers with Generalized Fibonacci and Lucas Numbers Coefficients. Math. Sci. Appl. E-Notes. 2020 Mar. 1;8(1):55-68. doi:10.36753/mathenot.621602