The Sequences of Fibonacci and Lucas for Quadratic Fields

  • PABLO LAM-ESTRADA Escuela Superior de F\'isica y Matem\'aticas, Departamento de Matem\'aticas, Instituto Polit\'ecnico Nacional (Unidad Zacatenco), CDMX, M\'exico.
  • MYRIAM ROSALIA MALDONADO-RAMIREZ Escuela Superior de F\'isica y Matem\'aticas, Departamento de Matem\'aticas, Instituto Polit\'ecnico Nacional (Unidad Zacatenco), CDMX, M\'exico.
  • JOSE LUIS LOPEZ-BONILLA Escuela Superior de Ingenier\'ia Mec\'anica y El\'ectrica, Departamento de Ingenier\'ia en Comunicaciones y Electr\'onica, Instituto Polit\'ecnico Nacional (Unidad Zacatenco), CDMX, M\'exico.
  • FAUSTO JARQUIN-ZARATE Universidad Aut\'onoma de la Ciudad de M\'exico, Academia de Matem\'aticas, Plantel San Lorenzo Tezonco, CDMX, M\'exico
  • R. RAJENDRA Department of Mathematics, Field Marshal K.M. Cariappa College (A Constituent College of Mangalore University), Madikeri-571 201, India
  • Siva Kota Reddy Polaepalli JSS Science and Technology University http://orcid.org/0000-0003-4033-8148

Resumen

We construct the sequences of Fibonacci and Lucas in any quadratic
field $\mathbb{Q}(\sqrt{d}\,)$ with $d>0$ square free, noting that
the general properties remain valid as those given by the
classical sequences of Fibonacci and Lucas for the case $d = 5$,
under the respective variants. For this construction, we use the
fundamental unit of $\mathbb{Q}(\sqrt{d}\,)$ and then we observe the
generalizations for any unit of $\mathbb{Q}(\sqrt{d}\,)$. Under
certain conditions some of these constructions correspond to
$k$-Fibonacci sequence for some $k\in \mathbb{N}$. Further, for
both sequences, we obtain the generating function, Golden ratio,
Binet's formula and some identities that they keep.

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Biografía del autor/a

Siva Kota Reddy Polaepalli, JSS Science and Technology University

Professor, Departmnet of Mathematics, JSS Science and Technology, Mysuru-570 006, India

Publicado
2025-01-21
Sección
Research Articles