Jacobi–Perron Algorithm and Recurrence Relations

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Da Fonseca C. M., Özkan E., Shannon A. G.

MATHEMATICS, cilt.14, sa.19, ss.1-17, 2026 (SCI-Expanded, Scopus)

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 14 Sayı: 19
  • Basım Tarihi: 2026
  • Doi Numarası: 10.3390/math14193495
  • Dergi Adı: MATHEMATICS
  • Derginin Tarandığı İndeksler: Academic Search Ultimate (EBSCO), Scopus, Materials Science & Engineering Collection (ProQuest), Technology Collection (ProQuest), Aerospace Database, Science Citation Index Expanded (SCI-EXPANDED), zbMATH, Directory of Open Access Journals
  • Sayfa Sayıları: ss.1-17
  • Açık Arşiv Koleksiyonu: AVESİS Açık Erişim Koleksiyonu
  • Marmara Üniversitesi Adresli: Evet

Özet

This paper explores connections between multidimensional continued fractions, specifically the Jacobi–Perron algorithm, and generalized Fibonacci and Lucas sequences of arbitrary order r. Recurrence relations are established, a sufficient criterion for the convergence of the associated ratio vectors is proved, and the limit is identified explicitly in terms of the dominant root of the auxiliary equation. When the coefficient of the oldest term of the recurrence equals one, this limit is shown to be the initial vector of a purely periodic Jacobi–Perron algorithm. The results link the work of Bernstein, Williams, and related authors to fundamental recursive sequences.