Turkish Journal of Mathematics
Abstract
This paper examines the sequences produced by the natural action of specific elements of the modular group on extended rational numbers. The polynomial sequences Pr(c) and Qr(c) are derived from the orbit of the point at infinity under a specific modular transformation. These sequences satisfy linear recurrence relations, which are analyzed using generating functions. The polynomials encode k-Fibonacci numbers, and studying their behavior modulo a fixed integer m reveals notable arithmetic and combinatorial properties. We also explore the connection between these modular actions and Farey graphs, illustrating the hyperbolic transformations as nested geodesic paths in the upper half-plane.
DOI
10.55730/1300-0098.3657
Keywords
Modular group action, recurrence relations, Fibonacci sequences
First Page
376
Last Page
389
Publisher
The Scientific and Technological Research Council of Türkiye (TÜBİTAK)
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.
Recommended Citation
KÖROĞLU, T, & GÜLER, B. Ö (2026). On sequences arising from the action of modular group. Turkish Journal of Mathematics 50 (3): 376-389. https://doi.org/10.55730/1300-0098.3657
