Inverses of Fibonacci and Lucas Numbers via Rational Indices
Fibonacci Quarterly, 2026 (ESCI, Scopus)
- Yayın Türü: Makale / Tam Makale
- Basım Tarihi: 2026
- Doi Numarası: 10.1080/00150517.2026.2638782
- Dergi Adı: Fibonacci Quarterly
- Derginin Tarandığı İndeksler: Emerging Sources Citation Index (ESCI), Scopus, MathSciNet, zbMATH
- Anahtar Kelimeler: Codenominator function, Fibonacci numbers, Lucas numbers
- Açık Arşiv Koleksiyonu: AVESİS Açık Erişim Koleksiyonu
- Bilecik Şeyh Edebali Üniversitesi Adresli: Evet
Özet
Fibonacci and Lucas numbers have been extensively studied in various algebraic and number-theoretic contexts, including their modular inverses and generalizations. Motivated by these developments, we study the inverses of Fibonacci and Lucas numbers with rational indices through the codenominator function (Formula presented.). Uludağ and Gökmen (2022) showed that the rational-indexed Fibonacci number (Formula presented.), where (Formula presented.), can be expressed using (Formula presented.), and that infinitely many such representations exist. In this paper, we extend their work by deriving a general explicit formula for (Formula presented.) through the codenominator function. We establish precise conditions under which (Formula presented.) coincides with a Lucas number or deviates from the classical Fibonacci sequence. Moreover, by means of these generalized formulas, we compute the units of Fibonacci and Lucas numbers, thereby proving the existence of multiplicative inverses for all Fibonacci and Lucas numbers within this framework. These results reveal new structural properties of Fibonacci- and Lucas-related sequences and suggest further directions for number-theoretic exploration.