On sequences arising from the action of modular group
TURKISH JOURNAL OF MATHEMATICS, cilt.50, sa.3, ss.376-389, 2026 (SCI-Expanded)
- Yayın Türü: Makale / Tam Makale
- Cilt numarası: 50 Sayı: 3
- Basım Tarihi: 2026
- Doi Numarası: 10.55730/1300-0098.3657
- Dergi Adı: TURKISH JOURNAL OF MATHEMATICS
- Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED)
- Sayfa Sayıları: ss.376-389
- Açık Arşiv Koleksiyonu: AVESİS Açık Erişim Koleksiyonu
- Karadeniz Teknik Üniversitesi Adresli: Evet
Özet
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.