The Complex-type Narayana-Fibonacci Numbers

Yıl 2023, Cilt: 13 Sayı: 1, 563 - 571, 01.03.2023

Yeşim Aküzüm

https://doi.org/10.21597/jist.1207287

Öz

In this paper, the complex-type Narayana-Fibonacci numbers are defined. Additionally, we arrive at correlations between the complex-type Narayana-Fibonacci numbers and this generating matrix after deriving the generating matrix for these numbers. Eventually, we get their the Binet formula, the combinatorial, permanental, determinantal, exponential representations, and the sums by matrix methods are just a few examples of numerous features.

Anahtar Kelimeler

The Narayana-Fibonacci number, matrix, representation

Kaynakça

• Adams, W. and Shanks, D. (1982). Strong Primality Tests that are not sufficient. Mathematics of Computation, 36(159), 255-300.
• Akuzum, Y. and Deveci, O. (2021). The arrowhead-Pell sequences. Ars Combinatoria, 155, 231-246.
• Akuzum, Y. and Deveci, O. (2022). The Narayana-Fibonacci sequence and its Binet formulas. 6th International Congress on Life, Social, and Health Sciences in a Changing World, İstanbul, July 02-03, pp:303-307.
• Becker, P.G. (1994). k-Regular power series and Mahler-type functional equations. Journal of Number Theory, 49(3), 269-286. Berzsenyi, G. (1975). Sums of products of generalized Fibonacci numbers. The Fibonacci Quarterly, 13(4), 43-344.
• Brualdi, R.A. and Gibson, P.M. (1977). Convex polyhedra of doubly stochastic matrices I: applications of permanent function. Journal of Combinatorial Theory, Series A, 22 (2), 194-230. Chen, W.Y.C. and Louck, J.D. (1996). The combinatorial power of the companion matrix. Linear Algebra and its Applications, 232, 261-278.
• Deveci, O. and Shannon, A.G. (2021). The complex-type k-Fibonacci sequences and their applications. Communications in Algebra, 49(3), 1352-1367.
• El Naschie, M.S. (2005). Deriving the Essential features of standard model from the general theory of relativity. Chaos, Solitons & Fractals, 26, 1-6. Erdag, O., Halici, S. and Deveci, O. (2022). The complex-type Padovan-p sequences. Mathematica Moravica, 26, 77-88.
• Fraenkel, A.S. and Klein, S.T. (1996). Robutst Universal complete codes for transmission and compression. Discrete Applied Mathematics, 64, 31-55.
• Gogin, N. and Myllari, A.A. (2007). The Fibonacci-Padovan sequence and MacWilliams transform matrices. Programming and Computer Software, 33, 74-79.
• Halici. S. and Deveci, O. (2021). On fibonacci quaternion matrix. Notes on Number Theory and Discrete Mathematics, 27(4), 236-244.
• Horadam, A.F. (1961). A generalized Fibonacci sequence. The American Mathematical Monthly, 68(5), 455-459.
• Horadam, A.F. (1963). Complex Fibonacci numbers and Fibonacci quaternions. The American Mathematical Monthly, 70(3), 289-291.
• Kalman, D. (1982). Generalized Fibonacci numbers by matrix methods. The Fibonacci Quarterly, 20, 73-76.
• Kirchoof, B.K. and Rutishauser, R. (1990). The Phyllotaxy of Costus (Costaceae). Botanical Gazette, 51(1), 88-105.
• Lee, G-Y. (2000). k-Lucas numbers and associated bipartite graphs. Linear Algebra and its Applications, 320, 51-61. Mandelbaum, D.M. (1972). Synchronization of codes by means of Kautz’s Fibonacci encoding. IEEE Transactions on Information Theory, 18(2), 281-285.
• Ozkan, E. (2007). On truncated Fibonacci sequences. Indian Journal of Pure and Applied Mathematics, 38(4), 241-251.
• Shannon, A.G. and Bernstein, L. (1973). The Jacobi-Perron algorithm and the algebra of recursive sequences. Bulletin of the Australian Mathematical Society, 8, 261-277.
• Shannon, A.G. and Horadam, A.F. (1994). Arrowhead curves in a tree of Pythagorean triples. Journal of Mathematical Education in Science and Technology, 25, 255-261.
• Spinadel, V.W. (2002). The Metallic Means Family and Forbidden Symmetries. International Journal of Mathematics, 2(3), 279-288.
• Stakhov, A. and Rozin, B. (2006). Theory of Binet formulas for Fibonacci and Lucas p-numbers. Chaos, Solitons & Fractals, 27, 1162-1177.
• Stein, W. (1993). Modelling The Evolution of Stelar Architecture in Vascular Plants. International Journal of Plant Sciences, 154(2), 229-263.
• Tasci, D. and Firengiz, M.C. (2010). Incomplete Fibonacci and Lucas p-numbers. Mathematical and Computer Modelling, 52(9-10), 1763-1770.
• Tuglu, N., Kocer, E.G. and Stakhov, A. (2011). Bivariate Fibonacci like p-polynomials. Applied Mathematics and Computation, 217, 10239-10246.
• Yılmaz, F. and Bozkurt, D. (2009). The generalized order-k Jacobsthal numbers. International Journal of Contemporary Mathematical Sciences, 4, 1685-1694.