**Orhan Dişkaya, Hamza Menken and Paula Maria Machado Cruz Catarino**

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 29, 2023, Number 3, Pages 407–420

DOI: 10.7546/nntdm.2023.29.3.407-420

**Full paper (PDF, 237 Kb)**

## Details

### Authors and affiliations

Orhan Dişkaya

*Department of Mathematics, Mersin University
Mersin, Turkey*

Hamza Menken

*Department of Mathematics, Mersin University
Mersin, Turkey*

Paula Maria Machado Cruz Catarino

*Department of Mathematics, University of Trás-os-Montes and Alto Douro
Vila Real, Portugal*

### Abstract

In this paper, we intruduce the bivariate Padovan sequence we examine its various identities. We define the bivariate Padovan polynomials matrix. Then, we find the Binet formula, generating function and exponential generating function of the bivariate Padovan polynomials matrix. Also, we obtain a sum formula and its series representation.

### Keywords

- Padovan numbers
- Padovan polynomials
- Binet-like formula
- Generating function
- Bivariare polynomials
- Matrix

### 2020 Mathematics Subject Classification

- 11B39
- 05A15

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### Manuscript history

- Received: 9 October 2022
- Revised: 24 March 2023
- Accepted: 28 May 2023
- Online First: 5 June 2023

### Copyright information

Ⓒ 2023 by the Authors.

This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

## Related papers

- Çakmak, T., & Karaduman, E. (2018). On the derivatives of bivariate Fibonacci polynomials.
*Notes on Number Theory Discrete Mathematics*, 24(3), 37–46. - Deveci, Ö., & Shannon, A. G. (2017). Pell–Padovan-circulant sequences and their applications.
*Notes on Number Theory and Discrete Mathematics*, 23(3), 100–114. - Dos Santos Mangueira, M. C., Vieira, R. P. M., Alves, F. R. V., & Catarino, P. M. M. C. (2022). Leonardo’s bivariate and complex polynomials.
*Notes on Number Theory and Discrete Mathematics*, 28(1), 115–123. - Vieira, R. P. M., dos Santos Mangueira, M. C., Alves, F. R. V., & Catarino, P. M. M. C. (2021). Perrin’s bivariate and complex polynomials.
*Notes on Number Theory and Discrete Mathematics*, 27(2), 70–78.

## Cite this paper

Dişkaya, O., Menken, H., & Catarino, P. M. M. C. (2023). On the bivariate Padovan polynomials matrix. *Notes on Number Theory and Discrete Mathematics*, 29(3), 407-420, DOI: 10.7546/nntdm.2023.29.3.407-420.