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

Year 2020, , 55 - 68, 20.03.2020

Arzu Cihan Ayşe Zeynep Azak , Mehmet Ali Güngör

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

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).
• [11] Yuce S. and Aydın Torunbalcı, F.: Generalized Dual Fibonacci Quaternions. Appl. Math. E-Notes 16, 276{289, (2016).