On second-order linear difference equations associated with hybrid sequences and generating functions

Authors

DOI:

https://doi.org/10.5902/2179460X84462

Keywords:

Hybrid sequences, Linear homogeneous recurrences, Binet's formula, Generating functions

Abstract

This article presents explicit formulas for the class of homogeneous linear recurrence of order 2 with constant coefficients associated with hybrid sequences, determined through generating functions. Furthermore, the applications of the resolution method in each case are all displayed and the relations between Binet's formulas and the expressions obtained via generating functions are discussed. Illustrative examples are given to clarify the approach.

Downloads

Download data is not yet available.

Author Biographies

Adriana Wagner, Universidade Federal de Mato Grosso do Sul

Bachelor's degree in Mathematics from the State University of Maringá (2004), Master's degree in Mathematics from the State University of Maringá (2008) and PhD in Applied Mathematics from the State University of Campinas (2016). She is currently an associate professor at the Federal University of Mato Grosso do Sul. She has experience in the area of ​​Mathematics, with an emphasis on Algebra and Combinatorics. Acting mainly on the following topics: Groups, Rings, Fields, Partitions and Number Theory.

Andreia Cristina Ribeiro, Universidade Federal de Mato Grosso do Sul

He holds a degree in Mathematics (Bachelor's Degree) from Universidade Estadual Paulista Júlio de Mesquita Filho (2000), a Master's degree in Mathematics from Universidade Estadual Paulista Júlio de Mesquita Filho (2003) and a PhD in Mathematics from Universidade Estadual de Campinas (2006). She worked as an effective professor (10/2006 to 02/2008) at the State University of Mato Grosso-UNEMAT. She is currently Associate Professor IV at the Federal University of the State of Mato Grosso do Sul - UFMS. He has experience in the area of Mathematics, with emphasis on Number Theory, working mainly on the following topics: principles of counting and partitions. He was in the direction of the Paranaíba Campus of UFMS from 2013 to 2021.

Elen Viviani Pereira Spreafico, Universidade Federal de Mato Grosso do Sul

She holds a degree in Mathematics from the Universidade Estadual Paulista Júlio de Mesquita Filho (2007), a master's degree (2010) and a doctorate (2014) in Applied Mathematics from the State University of Campinas. She is currently a professor at the Federal University of Mato Grosso do Sul, Campus Campo Grande - MS. She has works in the area of ​​Mathematics, with emphasis on Applied Mathematics, working mainly on Additive Number Theory, Discrete Mathematics and Combinatorics.

Irene Magalhães Craveiro, Universidade Federal da Grande Dourados

She is currently a professor at the Federal University of Grande Dourados. He has experience in the area of ​​Mathematics, with emphasis on Digital Modulations matched to groups, Discrete Mathematics and Combinatorics.

Tatiana Bertoldi Carlos, Universidade Federal de Mato Grosso do Sul

She holds a degree in Mathematics from the Paulista State University Júlio de Mesquita Filho (2000), a Master's degree in Mathematics from the Paulista State University Júlio de Mesquita Filho (2003) and a PhD in Mathematics from the State University of Campinas (2007). She is currently Associate Professor IV at the Federal University of Mato Grosso do Sul-UFMS. She has experience in Mathematics, with an emphasis on Algebraic Number Theory.

References

Catarino, P. M. M. (2019). On k-pell hybrid numbers. Journal of Discrete Mathematical Sciences and Cryptography, 22:1–7

Craveiro, I. M., Spreafico, E. V. P., & Rachidi, M. (2022). Recorrˆencias lineares homogˆeneas de ordem 2 com coeficientes constantes via func¸ ˜oes geradoras ordin´arias. ReviSeM, (1):86–106. DOI: https://doi.org/10.34179/revisem.v7i1.16274.

Lima, E. L. (2009). Curso de An´alise, Volume 1. Rio de Janeiro: IMPA, 12 edition.

Morales, G. C. (2018). Investigation of generalized hybrid Fibonacci numbers and their properties.

Szynal-Liana, A. (2018). The horadam hybrid numbers. Discussiones Mathematica e

General Algebra and Applications, 38(1):91–98.

Szynal-Liana, A. & Wlock, I. (2018). On pell and pell-lucas hybrid numbers. Commentat Math, (110):11–17. DOI: https://doi.org/10.14708/cm.v58i1-2.6364.

Szynal-Liana, A. & Wlock, I. (2019a). The fibonacci hybridnumbers. Utilitas Math, .(110):3–10. Available in: https://ulilitasmathematica.com/index.php/Index/article/view/1428.

Szynal-Liana, A. & Wlock, I. (2019b). On jacobsthal and jacobsthal-lucas hybrid numbers. Annales Mathematicae Silesianae, 33(1):276–283. DOI: https://doi.org/10.2478/amsil-2018-0009.

¨Ozdemir, M. (2018). Introduction to hybrid numbers. Advances in Applied Clifford Algebras, 28.

Downloads

Published

2025-01-29

How to Cite

Wagner, A., Ribeiro, A. C., Spreafico, E. V. P., Craveiro, I. M., & Carlos, T. B. (2025). On second-order linear difference equations associated with hybrid sequences and generating functions. Ciência E Natura, 46. https://doi.org/10.5902/2179460X84462

Issue

Vol. 46 (2024): Publicação contínua

Section

Mathematics

License

Copyright (c) 2024 Ciência e Natura

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

To access the DECLARATION AND TRANSFER OF COPYRIGHT AUTHOR’S DECLARATION AND COPYRIGHT LICENSE click here.

 

Ethical Guidelines for Journal Publication

The Ciência e Natura journal is committed to ensuring ethics in publication and quality of articles. 

Conformance to standards of ethical behavior is therefore expected of all parties involved: Authors, Editors, Reviewers, and the Publisher.

In particular, 

Authors: Authors should present an objective discussion of the significance of research work as well as sufficient detail and references to permit others to replicate the experiments. Fraudulent or knowingly inaccurate statements constitute unethical behavior and are unacceptable. Review Articles should also be objective, comprehensive, and accurate accounts of the state of the art. The Authors should ensure that their work is entirely original works, and if the work and/or words of others have been used, this has been appropriately acknowledged. Plagiarism in all its forms constitutes unethical publishing behavior and is unacceptable. Submitting the same manuscript to more than one journal concurrently constitutes unethical publishing behavior and is unacceptable. Authors should not submit articles describing essentially the same research to more than one journal. The corresponding Author should ensure that there is a full consensus of all Co-authors in approving the final version of the paper and its submission for publication.

Editors: Editors should evaluate manuscripts exclusively on the basis of their academic merit. An Editor must not use unpublished information in the editor's own research without the express written consent of the Author. Editors should take reasonable responsive measures when ethical complaints have been presented concerning a submitted manuscript or published paper.

Reviewers: Any manuscripts received for review must be treated as confidential documents. Privileged information or ideas obtained through peer review must be kept confidential and not used for personal advantage. Reviewers should be conducted objectively, and observations should be formulated clearly with supporting arguments, so that Authors can use them for improving the paper. Any selected Reviewer who feels unqualified to review the research reported in a manuscript or knows that its prompt review will be impossible should notify the Editor and excuse himself from the review process. Reviewers should not consider manuscripts in which they have conflicts of interest resulting from competitive, collaborative, or other relationships or connections with any of the authors, companies, or institutions connected to the papers.