Pell Leonardo numbers and their matrix representations

Year 2024, Volume: 13 Issue: 2, 101 - 108, 31.08.2024

Çağla Çelemoğlu

https://doi.org/10.54187/jnrs.1506171

Abstract

In this study, we investigate Pell numbers and Leonardo numbers and describe a new third-order number sequence entitled Pell Leonardo numbers. We then construct some identities, including the Binet formula, generating function, exponential generating function, Catalan, Cassini, and d’Ocagne’s identities for Pell Leonardo numbers and obtain a relation between Pell Leonardo and Pell numbers. In addition, we present some summation formulas of Pell Leonardo numbers based on Pell numbers. Finally, we create a matrix formula for Pell Leonardo numbers and obtain the determinant of the matrix.

Keywords

Leonardo numbers, Pell numbers, Binet’s formula, Generating function, Matrix representation

Ethical Statement

No approval from the Board of Ethics is required.

References

• M. Bicknell, A primer of the Pell sequence and related sequences, The Fibonacci Quarterly 13 (4) (1975) 345–349.
• A. F. Horadam, Applications of modified Pell numbers to representations, Ulam Quarterly 3 (1) (1994) 34-53.
• R. Melham, Sums involving Fibonacci and Pell numbers, Portugaliae Mathematica 56 (3) (1999) 309-318.
• S. F. Santana, J. L. Díaz-Barrero, Some properties of sums involving Pell numbers, Missouri Journal of Mathematical Sciences 18 (1) (2006) 33-40.
• Q. Mushtaq, U. Hayat, Pell numbers, Pell–Lucas numbers and modular group, In Algebra Colloquium 14 (1) (2007) 97-102.
• A. Dasdemir, On the Pell, Pell-Lucas and modified Pell numbers by matrix method, Applied Mathematical Sciences 5 (64) (2011) 3173-3181.
• S. Çelik, İ. Durukan, E. Özkan, New recurrences on Pell numbers, Pell-Lucas numbers, Jacobsthal numbers and Jacobsthal-Lucas numbers, Chaos, Solitons & Fractals 150 (2021), Article Number 111173 8 pages.
• P. M. Catarino and A. Borges, On Leonardo numbers, Acta Mathematica Universitatis Comenianae 89 (1) (2019) 75-86.
• A. G. Shannon, A note on generalized Leonardo numbers, Notes on Number Theory and Discrete Mathematics 25 (3) (2019) 97-101.
• Y. Alp, E. G. Koçer, Some properties of Leonardo numbers, Konuralp Journal of Mathematics 9 (1) (2021) 183-189.
• Y. Alp, E. G. Kocer, Hybrid Leonardo numbers, Chaos, Solitons & Fractals 150 (2021) Article Number 111128 5 pages.
• K. Kuhapatanakul, J. Chobsorn, On the generalized Leonardo Numbers, Integers: Electronic Journal of Combinatorial Number Theory 22 (2022) Article Number A48 7 pages.
• A. Karatas, On complex Leonardo numbers, Notes on Number Theory and Discrete Mathematics 28 (3) (2022) 458-465.
• E. Tan, H. H. Leung, On Leonardo p-numbers, Integers: Electronic Journal of Combinatorial Number Theory 23 (2023) 1-11.
• Y. Soykan, Generalized Horadam-Leonardo numbers and polynomials, Asian Journal of Advanced Research and Reports 17 (8) (2023) 128-169.