On Various Binomial Transforms of the Generalized Pell Matrix Sequences

Abstract

The main aim of this study is to apply different binomial transforms to the generalized Pell matrix sequences. We define the binomial, $s$-binomial, rising, and falling transforms for the generalized Pell matrix sequence. We establish some algebraic properties, such as the recurrent formulas, Binet formulas, generating functions, and sum formulas, for binomial transforms of the generalized Pell matrix sequences. Finally, we discuss the need for further research.

Keywords

• Binomial transforms
• matrix sequences
• generating function
• Binet formula
• Pell numbers

References

1. H. H. Gulec , N. Taskara, On the $(s,t)$-Pell and $(s,t)$-Pell-Lucas sequences and their matrix representations, Applied Mathematics Letters 25 (10) (2012) 1554–1559.
2. S. Uygun, Z. S. Açar, Notes on $(s,t)$-Pell and $(s,t)$-Pell Lucas matrix sequences, Asian Journal of Mathematics and Physics 7 (2023) 1.
3. A. F. Horadam, Pell identities, Fibonacci Quarterly 9 (3) (1971) 245–252.
4. T. Koshy, Pell and Pell-Lucas numbers with applications, Springer, 2014.
5. H. Prodinger, Some information about the binomial transform, Fibonacci Quarterly 32 (5) (1994) 412–415.
6. K. W. Chen, Identities from the binomial transform, Journal of Number Theory 124 (1) (2007) 142–150.
7. S. Falcon, A. Plaza, Binomial transforms of the $k$-Fibonacci sequence, International Journal of Nonlinear Sciences & Numerical Simulation 10 (11-12) (2009) 1527–1538.
8. P. Bhadouria, D. Jhala, B. Singh, Binomial transforms of the $k$-Lucas sequences and its properties, Journal of Mathematics and Computer Science 8 (2014) 81–92.

9. N. Yilmaz, N. Taskara, Binomial transforms of the Padovan and Perrin matrix sequences, Abstract and Applied Analysis 2013 (1) 497418.
10. S. Uygun, A. Erdoğdu, Binomial transforms of $k$-Jacobsthal sequences, Journal of Mathematical and Computational Science 7 (6) (2017) 1100–1114.
11. C. Kızılateş, N. Tuglu, B. Çekim, Binomial transform of quadrapell sequences and quadrapell matrix sequences, Journal of Science and Arts 1 (38) (2017) 69–80.
12. S. Uygun, The binomial transforms of the generalized $(s,t)$-Jacobsthal matrix sequence, International Journal of Advances in Applied Mathematics and Mechanics 6 (3) (2019) 14–20.
13. F. Kaplan, A. Özkoç Öztürk, On the binomial transforms of the Horadam quaternion sequences, Mathematical Methods in the Applied Sciences 45 (18) (2021) 12009–12022.
14. Y. Kwon, Binomial transforms of the modified $k$-Fibonacci-like sequence, International Journal of Mathematics and Computer Science 14 (1) (2019) 47–59.
15. Y. Soykan, Binomial transform of the generalized Tribonacci sequence, Asian Research Journal of Mathematics 16 (10) (2020) 26–55.
16. Y. Soykan, Binomial transform of the generalized third order Pell sequence, Communications in Mathematics and Applications 12 (1) (2021) 71–94.
17. Y. Soykan, Binomial transform of the generalized fourth order Pell sequence, Archives of Current Research International 21 (6) (2021) 9–31.
18. Y. Soykan, On binomial transform of the generalized fifth order Pell sequence, Asian Journal of Advanced Research and Reports 15 (9) (2021) 8–29.
19. Y. Soykan, Notes on binomial transform of the generalized Narayana sequence, Earthline Journal of Mathematical Sciences 7 (1) (2021) 77–111.
20. Y. Soykan, Binomial transform of the generalized Pentanacci sequence, Asian Research Journal of Current Science 3 (1) (2021) 209–231.
21. Y. Soykan, E. Taşdemir, N. Özmen, On binomial transform of the generalized Jacobsthal-Padovan numbers, International Journal of Nonlinear Analysis and Applications 14 (1) (2021) 643–666.