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New Results on Negative-Indexed Pell Numbers via Matrix Methods

Year 2025, Volume: 46 Issue: 4, 902 - 910, 30.12.2025

İbrahim Gökcan , Ümmügülsün Çağlayan , Ali Hikmet Değer

https://doi.org/10.17776/csj.1674240

Abstract

In this study, we investigate the Pell and Pell–Lucas numbers sequences and construct matrices whose elements are defined using negative indices of these sequences through Binet’s formula. Identities involving negative-indexed Pell and Pell–Lucas numbers are derived using the determinants and traces of these matrices. Furthermore, we examine various powers of the constructed matrices and discuss related elementary matrix operations.

Keywords

Pell numbers Pell-Lucas numbers matrix theory

References

  • [1] Taş N., Uçar S., Yılmaz Özgür N., Pell coding and Pell decoding methods with some applications, Contrib. Discrete Math, 15(1) (2020) 52-66.
  • [2] Jiang Z., Li J.,Shen N. , On explicit determinants of RFPLR and RFPLL circulant matrices involving Pell numbers in information theory,International Conference on Information Computing and Applications, (2012) 364-370.
  • [3] Dasdemir A., On the Pell, Pell-Lucas and modified Pell numbers by matrix method,Applied Mathematical Sciences, 5(64)(2011)3173-3181.
  • [4] Horadam A., Applications of Modified Pell Numbers to Representations,Ulam Quarterly, 3(1)( 1994) 34-53.
  • [5] Horadam A., Pell identities,Fibonacci Quarterly, 9(3)(1971) 245-252.
  • [6] Kilic E., Tasci D., The linear algebra of the Pell matrix,Bolet´ın de la Sociedad Matem´atica Mexicana, 11(3)(2005)163-174.
  • [7] Ercolano J.,Matrix generators of Pell sequence,Fibonacci Quarterly, 17(1)(1979) 71-77.
  • [8] Akbaba U., Deger A., Relation between matrices and the suborbital graphs by the special number sequences,Turkish Journal of Mathematics, 42 (2022) 753-767.
  • [9] Akbaba U., Değer A., On applications of Pell and Pell-Lucas numbers with matrix method,Journal of Intelligent Fuzzy Systems, 44(6)( 2023) 10703-10707.
  • [10] Koshy T., Pell and Pell-Lucas Numbers with Applications, New York: Springer, (2014).
  • [11] Yaşar M., Bozkurt D., Negatively Indexed Pell Numbers as the Permanent of Tridiagonal Matrix,Palestine Journal of Mathematics, 5(1)(2016)147-150.
  • [12] Kannika K., Sompong S., On matrix sequences represented by negative indices Pell and Pell–Lucas number with the decoding of Lucas blocking error correcting codes,Discrete Mathematics, Algorithms and Applications, 15(07)(2023)2250154.
  • [13] Nabiha S., Boussayoud A., Ordinary generating functions of binary products of (p, q)-modified Pell numbers and k-numbers at positive and negative indices,Journal of Science and Arts, 20( 3) ( 2020)627-646.
  • [14] Kishore J.,Verma V., Some representations of sums of finite products of Fibonacci type numbers and polynomials,Journal of Information and Optimization Sciences, 46(7)( 2025) 2215-2229.
  • [15] Garcia J., Gomez C.A., Luca F., Identities for the k-generalized Fibonacci sequence with negative indices and its zero multicipty,arXiv preprint arXiv:2211.00248(2022).
  • [16] Soykan Y., Okumuş İ., Göcen M., On Generalized Tetranacci Quaternions,Sohag Journal of Mathematics, 10(2)( 2023) 11-18.

Year 2025, Volume: 46 Issue: 4, 902 - 910, 30.12.2025

İbrahim Gökcan , Ümmügülsün Çağlayan , Ali Hikmet Değer

https://doi.org/10.17776/csj.1674240

Abstract

References

  • [1] Taş N., Uçar S., Yılmaz Özgür N., Pell coding and Pell decoding methods with some applications, Contrib. Discrete Math, 15(1) (2020) 52-66.
  • [2] Jiang Z., Li J.,Shen N. , On explicit determinants of RFPLR and RFPLL circulant matrices involving Pell numbers in information theory,International Conference on Information Computing and Applications, (2012) 364-370.
  • [3] Dasdemir A., On the Pell, Pell-Lucas and modified Pell numbers by matrix method,Applied Mathematical Sciences, 5(64)(2011)3173-3181.
  • [4] Horadam A., Applications of Modified Pell Numbers to Representations,Ulam Quarterly, 3(1)( 1994) 34-53.
  • [5] Horadam A., Pell identities,Fibonacci Quarterly, 9(3)(1971) 245-252.
  • [6] Kilic E., Tasci D., The linear algebra of the Pell matrix,Bolet´ın de la Sociedad Matem´atica Mexicana, 11(3)(2005)163-174.
  • [7] Ercolano J.,Matrix generators of Pell sequence,Fibonacci Quarterly, 17(1)(1979) 71-77.
  • [8] Akbaba U., Deger A., Relation between matrices and the suborbital graphs by the special number sequences,Turkish Journal of Mathematics, 42 (2022) 753-767.
  • [9] Akbaba U., Değer A., On applications of Pell and Pell-Lucas numbers with matrix method,Journal of Intelligent Fuzzy Systems, 44(6)( 2023) 10703-10707.
  • [10] Koshy T., Pell and Pell-Lucas Numbers with Applications, New York: Springer, (2014).
  • [11] Yaşar M., Bozkurt D., Negatively Indexed Pell Numbers as the Permanent of Tridiagonal Matrix,Palestine Journal of Mathematics, 5(1)(2016)147-150.
  • [12] Kannika K., Sompong S., On matrix sequences represented by negative indices Pell and Pell–Lucas number with the decoding of Lucas blocking error correcting codes,Discrete Mathematics, Algorithms and Applications, 15(07)(2023)2250154.
  • [13] Nabiha S., Boussayoud A., Ordinary generating functions of binary products of (p, q)-modified Pell numbers and k-numbers at positive and negative indices,Journal of Science and Arts, 20( 3) ( 2020)627-646.
  • [14] Kishore J.,Verma V., Some representations of sums of finite products of Fibonacci type numbers and polynomials,Journal of Information and Optimization Sciences, 46(7)( 2025) 2215-2229.
  • [15] Garcia J., Gomez C.A., Luca F., Identities for the k-generalized Fibonacci sequence with negative indices and its zero multicipty,arXiv preprint arXiv:2211.00248(2022).
  • [16] Soykan Y., Okumuş İ., Göcen M., On Generalized Tetranacci Quaternions,Sohag Journal of Mathematics, 10(2)( 2023) 11-18.
There are 16 citations in total.

Details

Primary Language English
Subjects Algebra and Number Theory
Journal Section Research Article
Authors

İbrahim Gökcan 0000-0002-6933-8494

Ümmügülsün Çağlayan 0000-0002-5870-6802

Ali Hikmet Değer 0000-0003-0764-715X

Submission Date April 11, 2025
Acceptance Date December 20, 2025
Publication Date December 30, 2025
Published in Issue Year 2025 Volume: 46 Issue: 4

Cite

APA Gökcan, İ., Çağlayan, Ü., & Değer, A. H. (2025). New Results on Negative-Indexed Pell Numbers via Matrix Methods. Cumhuriyet Science Journal, 46(4), 902-910. https://doi.org/10.17776/csj.1674240

As of 2026, Cumhuriyet Science Journal will be published in six issues per year, released in February, April, June, August, October, and December