Some inclusion results for the new Tribonacci-Lucas matrix

Abstract

The main purpose of this paper is first to establish a new regular matrix by using one of the important sequences of integer number called Tribonacci-Lucas. Also, we class this new Tribonacci-Lucas matrix with some well-known summability methods such as Riesz means, Nörlund means and Cesaro means. To do this, we show that the Tribonacci-Lucas matrix is a regular summability method and in addition to this, we give some inclusion results and finally prove that Cesaro matrix is stronger than the Tribonacci-Lucas matrix.

Keywords

• Tribonacci-Lucas numbers
• Fibonacci numbers
• Toeplitz matrix
• Summability method

References

1. [1] Feinberg, M., 1963. Fibonacci-Tribonacci. The Fibonacci Quarterly, 1 (3), 70-74.
2. [2] Catalani, M., 2002. Identities for Tribonacci-related Sequences, Cornell University Library, arXiv:0209179.
3. [3] Frontczak, R., 2018. Sums of Tribonacci and Tribonacci-Lucas Numbers, International Journal of Mathematical Analysis, 12 (1), 19-24.
4. [4] Kılıç, E., 2008. Tribonacci Sequences with Certain Indices and Their Sums, Ars Combinatoria, 86, 13-22.
5. [5] Kara, E.E., and Başarır, M., 2012. An Application of Fibonacci Numbers into Infinite Toeplitz Matrices, Caspian Journal of Mathematical Sciences, 1 (1), 43-47.
6. [6] Karakaş, M., and Karakaş, A.M., 2017. New Banach Sequence Spaces That is Defined By The Aid of Lucas Numbers, Iğdır University Journal of the Institute of Science and Technology, 7 (4), 103-111.
7. [7] Karakaş, M., and Karakaş, A.M., 2018. A Study on Lucas Difference Sequence Spaces and , Maejo International Journal of Science and Technology, 12 (1), 70-78.
8. [8] Yaying, T., Hazarika, B., and Mohiuddine, S.A., 2021. On Difference Sequence Spaces of Fractional Order Involving Padovan Numbers, Asian-European Journal of Mathematics, 14 (6), 1-24.

9. [9] Yaying, T., and Hazarika, B., 2020. On Sequence Spaces Defined by the Domain of a Regular Tribonacci Matrix, Mathematica Slovaca, 70 (3), 697-706.
10. [10] Ercan, S., and Bektaş, Ç., 2017. Some Topological and Geometric Properties of a New BK-space Derived by Using Regular Matrix of Fibonacci Numbers, Linear and Multilinear Algebra, 65 (5), 909-921.
11. [11] Petersen, G.M., 1966. Regular Matrix Transformations. McGraw Hill Publishing Company Limited.
12. [12] Başar, F., 2011. Summability Theory and Its Applications. Bentham Science Publishers.
13. [13] Armitage, D.H., and Maddox, I.J., 1989. A New Type of Cesaro Mean, Analysis, 9, 195-204.
14. [14] Küçükaslan, M., and Aris, B., 2019. Sequence Spaces Defined by Fibonacci Matrix, General Letters in Mathematics, 6 (2), 45-60.