Some approximations and identities from special sequences for the vertices of suborbital graphs

Yıl 2024, Cilt: 42 Sayı: 5, 1439 - 1447, 04.10.2024

İbrahim Gökcan , Ali Hikmet Değer , Ümmügülsün Çağlayan

Öz

In this study, we investigate the vertices arising from the action of a suborbital graph, in terms of continued fractions, matrix, and recurrence relations. Using the approximation of Fibo-nacci sequence by the Binet formula, we demonstrate that the vertices of the suborbital graph are related to Lucas numbers. Then, we provide new identities and approximations regarding Fibonacci, Lucas, Pell, and Pell-Lucas numbers.

Anahtar Kelimeler

Continued Fraction, Generalized Fibonacci Numbers, Modular Group

Kaynakça

• REFERENCES
• [1] Sims CC. Graphs and finite permutation groups. Math. Z 1967;95:7686. [CrossRef]
• [2] Jones GA, Singerman D, Wicks K. The modular group and generalized Farey graphs. London Math Soc Lecture Note Ser 1991;160:316338. [CrossRef]
• [3] Akbas M. On suborbital graphs for modular group. Bull Lond Math Soc 2001;33:647652. [CrossRef]
• [4] Deger AH, Besenk M, Guler BO, On suborbital graphs and related continued fractions. Appl. Math. Comput 2011;218:746750. [CrossRef]
• [5] Deger AH, Relationships with the Fibonacci numbers and the special vertices of the suborbital graphs. GUJS 2017;7:168180.
• [6] Deger AH, Vertices of paths of minimal lengths on suborbital graphs. Filomat 2017;31:913923. [CrossRef]
• [7] Woschni EG, The importance of estimation and approximation methods in system theory. Cybern Syst 1992;23:335343. [CrossRef]
• [8] Koshy T, Fibonacci and Lucas Numbers with Applications. New York: Wiley-Interscience Publication; 2001. [CrossRef]
• [9] Ratcliffe JG. Foundations of Hyperbolic Manifolds. New York: Springer-Verlag; 1994. [CrossRef]
• [10] Schoeneberg B. Elliptic Modular Functions. New York: Springer-Verlag; 1974. [CrossRef]
• [11] Anderson JW. Hyperbolic Geometry. Southampton: Springer; 2005.
• [12] Diestel R. Graph Theory. New York: Springer-Verlag Heidelberg; 2005. [CrossRef]
• [13] Ruohonen K. Graph Theory. Tampere, Finland: Tampere University of Technology; 2008.
• [14] Neumann PM, Finite Permutation Groups, Edge Coloured Graphs and Matrices, Topics in Group Theory and Computation. Curran M.P.J. (eds.) London: Academic Press; 1977.
• [15] Tsukuzu T. Finite Groups and Finite Geometries. Cambridge: Cambridge University Press; 1982.
• [16] Akbaba U, Deger AH, Relation between matrices and the suborbital graphs by the special number sequences. Turkish J Math 2022;46:753767. [CrossRef]
• [17] Kişi Ö, Debnath P, Fibonacci ideal convergence on intuitionistic fuzzy normed linear spaces. Fuzzy Inform Eng 2022;14:255268. [CrossRef]
• [18] Kişi Ö, Fibonacci lacunary ideal convergence of double sequences in intuitionistic fuzzy normed linear spaces. Math Sci Appl E-Notes 2022;10:114124. [CrossRef]