Merve Taştan and Engin Özkan

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 27, 2021, Number 1, Pages 198—207

DOI: 10.7546/nntdm.2021.27.1.198-207

**Download full paper: PDF, 865 Kb**

## Details

### Authors and affiliations

Merve Taştan

*Graduate School of Natural and Applied Sciences
Erzincan Binali Yıldırım University
24100 Erzincan, Turkey*

Engin Özkan

*Graduate School of Natural and Applied Sciences
Erzincan Binali Yıldırım University
24100 Erzincan, Turkey*

### Abstract

In this study, we present the Catalan transforms of the *k*-Pell sequence, the *k*-Pell–Lucas sequence and the Modified *k*-Pell sequence and examine the properties of the sequences. Then we apply the Hankel transform to the Catalan transforms of the *k*-Pell sequence, the Catalan transform of the *k*-Pell–Lucas sequence and the Catalan transform of the Modified *k*-Pell sequence. Also, we obtain the generating functions of the Catalan transform of the *k*-Pell sequence, *k*-Pell–Lucas sequence and Modified *k*-Pell sequence. Furthermore, we acquire an interesting characteristic related to the determinant of the Hankel transform of the sequences.

### Keywords

*k*-Pell numbers*k*-Pell–Lucas- Catalan numbers
- Catalan transform
- Hankel transform

### 2010 Mathematics Subject Classification

- 11B39
- 11B83
- 11C20

### References

- Barry, P. (2005). A Catalan transform and related transformations on integer sequences. Journal of Integer Sequences, 8(4), 1–24.
- Catarino, P., & Campos, H. (2017). Incomplete
*k*-Pell,*k*-Pell–Lucas and Modified*k*-Pell numbers. Hacettepe Journal of Mathematics and Statistics, 46 (3), 361–372. - Catarino, P., & Vasco, P. (2013). On Some Identities and Generating Functions for k-Pell–Lucas Sequence. Applied Mathematical Sciences, 7(98), 4867–4873.
- Catarino, P. (2013). On some identities and generating functions for
*k*-Pell numbers. International Journal of Mathematical Analysis, 7(38), 1877–1884. - Cvetković, A., Rajković, R., & Ivković, M. (2002). Catalan numbers, and Hankel transform, and Fibonacci numbers. Journal of Integer Sequences, 5(1), 1–8.
- Deveci, Ö., & Shannon, A. G. (2018). The quaternion-Pell sequence. Communications in Algebra, 46(12), 5403–5409.
- Falcon, S. (2013). Catalan Transform of the
*k*-Fibonacci sequence. Communications of the Korean Mathematical Society, 28(4), 827–832. - Horadam, A. F. (1971). Pell identities. The Fibonacci Quarterly, 9(3), 245–263.
- Layman, J. W. (2001). The Hankel transform and some of its properties. Journal of Integer Sequences, 4(1), Article 01.1.5.
- Marin, M. (1997). On the domain of influence thermoelasticity of bodies with voids. Archivum Mathematicum, 33(4), 301–308.
- Özkan, E. (2003). 3-Step Fibonacci Sequences in Nilpotent Groups. Applied Mathematics and Computation, 144, 517–527.
- Özkan, E., & Taştan, M. (2019). On Gauss Fibonacci Polynomials, Gauss Lucas Polynomials and Their Applications. Communications in Algebra, 48(3), 952–960.
- Özkan, E., Taştan, M., & Aydoğdu, A. (2018). 2-Fibonacci Polynomials in the Family of Fibonacci Numbers. Notes on Number Theory and Discrete Mathematics, 24(3), 47–55.
- Özkan, E., Taştan, M., & Güngör, O. (2020). Catalan Transform of the k-Lucas Numbers. Erzincan University Journal of Science and Technology, 13, 145–149.
- Rajković, P. M., Petković, M. D., & Barry, P. (2007). The Hankel transform of the sum of consecutive generalized Catalan numbers. Integral Transforms and Special Functions, 18(4), 285–296.
- Sloane, N. J. A. (2013). The on-line encyclopedia of integer sequences. Annales Mathematicae et Informaticae, 41, 219–234.
- Taştan, M. & Özkan, E. (2020). Catalan Transform of the
*k*-Jacobsthal Sequence. Electronic Journal of Mathematical Analysis and Applications,8(2), 70–74. - Wikipedia contributors. (2020, May 7). Catalan number. In Wikipedia, The Free Encyclopedia, from https://en.wikipedia.org/w/index.php?title=Catalan_number&oldid=955372145.

## Related papers

## Cite this paper

Taştan, M. & Özkan, E. (2021). Catalan transform of the *k*-Pell, *k*-Pell–Lucas and modified *k*-Pell sequence. Notes on Number Theory and Discrete Mathematics, 27(1), 198-207, doi: 10.7546/nntdm.2021.27.1.198-207.