Cahit Köme

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 27, 2021, Number 4, Pages 207–218

DOI: 10.7546/nntdm.2021.27.4.207-218

**Full paper (PDF, 264 Kb)**

## Details

### Authors and affiliations

Cahit Köme

*Department of Information Technology,
Nevşehir Hacı Bektaş Veli University
50300, Turkey*

### Abstract

In this study, we investigate the connection between second order recurrence matrix and several combinatorial matrices such as generalized *r*-eliminated Pascal matrix, Stirling matrix of the first and of the second kind matrices. We give factorizations and inverse factorizations of these matrices by virtue of the second order recurrence matrix. Moreover, we derive several combinatorial identities which are more general results of some earlier works.

### Keywords

- Pascal matrix
- Stirling matrix
- Second order recurrence matrix
- Factorization
- Combinatorial identities

### 2020 Mathematics Subject Classification

- 15A23
- 11B39
- 05A10

### References

- Aceto, L., & Trigiante, D. (2001). The matrices of Pascal and other greats. The American Mathematical Monthly, 108(3), 232–245.
- Bayat, M., & Teimoori, H. (2000). Pascal k-eliminated functional matrix and its property. Linear Algebra and Its Applications, 308(1–3), 65–75.
- Brawer, R., & Pirovino, M. (1992). The linear algebra of the Pascal matrix. Linear Algebra and Its Applications, 174, 13–23.
- Brualdi, R. A. (1977). Introductory Combinatorics. Pearson Education India.
- Cheon, G. S., & Kim, J. S. (2001). Stirling matrix via Pascal matrix. Linear Algebra and its Applications, 329(1–3), 49–59.
- Comtet, L. (2012). Advanced Combinatorics: The Art of Finite and Infinite Expansions. Springer Science & Business Media.
- Irmak, N., & Köme, C. (2021). Linear Algebra of the Lucas Matrix. Hacettepe Journal of Mathematics and Statistics, 50(2), 549–558.
- Lee, G. Y., & Cho, S. H. (2008). The generalized Pascal matrix via the generalized Fibonacci matrix and the generalized Pell matrix. Journal of the Korean Mathematical Society, 45(2), 479–491.
- Lee, G. Y., Kim, J. S., & Cho, S. H. (2003). Some combinatorial identities via Fibonacci numbers. Discrete Applied Mathematics, 130(3), 527–534.
- Quintana, Y., Ramírez, W., & Urieles G., A. (2019). Generalized Apostol-type polynomial matrix and its algebraic properties. Mathematical Reports, 21(71), 2, 249–264.
- Stanica, P. (2005). Cholesky factorizations of matrices associated with r-order recurrent sequences. Integers: Electronic Journal of Combinatorial Number Theory, 5(2), A16.
- Stanimirovic, P., Nikolov, J., & Stanimirovic, I. (2008). A generalization of Fibonacci and Lucas matrices. Discrete Applied Mathematics, 156(14), 2606–2619.
- Zhang, Z. (1997). The linear algebra of the generalized Pascal matrix. Linear Algebra and Its Applications, 250, 51–60.
- Zhang, Z., & Wang, X. (2007). A factorization of the symmetric Pascal matrix involving the Fibonacci matrix. Discrete Applied Mathematics, 155(17), 2371–2376.
- Zhang, Z., & Zhang, Y. (2007). The Lucas matrix and some combinatorial identities. Indian Journal of Pure and Applied Mathematics, 38(5), 457–465.

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## Cite this paper

Köme, C. (2021). Factorizations of some lower triangular matrices and related combinatorial identities. *Notes on Number Theory and Discrete Mathematics*, 27(4), 207-218, DOI: 10.7546/nntdm.2021.27.4.207-218.