Bahar Kuloğlu, Engin Özkan and Anthony G. Shannon

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 27, 2021, Number 4, Pages 245–256

DOI: 10.7546/nntdm.2021.27.4.245-256

**Full paper (PDF, 149 Kb)**

## Details

### Authors and affiliations

Bahar Kuloğlu

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

Engin Özkan

*Department of Mathematics, Faculty of Arts and Sciences
Erzincan Binali Yıldırım University, Erzincan, Turkey*

Anthony G. Shannon

*Warrane College, The University of New South Wales
Kensington 2033, Australia*

### Abstract

In this paper, we introduce the incomplete Vieta–Pell and Vieta–Pell–Lucas polynomials. We give some properties, the recurrence relations and the generating function of these polynomials with suggestions for further research.

### Keywords

- Binet’s formula
- Generating function
- Incomplete generalized Vieta–Pell polynomials
- Incomplete generalized Vieta–Pell–Lucas polynomials

### 2020 Mathematics Subject Classification

- 11B39
- 11B83

### References

- 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. - Dikici, R., & Özkan, E. (2003). An application of Fibonacci sequences in groups. Applied Mathematics and Computation, 136 (2–3), 323–331.
- Filipponi, P. (1996). Incomplete Fibonacci and Lucas numbers. Rendiconti del Circolo Matematico di Palermo, 45(2), 37–56.
- Filipponi, P., & Horadam, A. F. (1999). Integration sequences of Jacobsthal and Jacobsthal–Lucas Polynomials, in Fredric T Howard (ed.), Applications of Fibonacci Numbers, Volume 8. Dordrecht: Kluwer, 129–139.
- Kim, D. S., & Kim, T. (2021). Degenerate Sheffer sequence and
*λ*-Sheffer sequence. Journal of Mathematical Analysis and Applications, 493(1), 124521 - Koshy, T. (2001). Fibonacci and Lucas Numbers with Applications, A Wiley-Interscience, New York.
- Özkan, E. (2003). 3-Step Fibonacci Sequences in Nilpotent Groups. Applied Mathematics and Computation, 144, 517–527.
- Özkan E. (2003). On General Fibonacci Sequences in Groups. Turkish Journal of Mathematics, 27(4), 525–537.
- Özkan, E., & Altun, İ. (2019). Generalized Lucas Polynomials and Relationships between the Fibonacci Polynomials and Lucas Polynomials. Communications in Algebra, 47, 10–12.
- Özkan, E., & Taştan, M. (2020). On Gauss Fibonacci Polynomials, Gauss Lucas Polynomials and Their Applications. Communications in Algebra, 48(3), 952–960.
- Pinter, A., & Srivastava, H. M. (1995). Generating Functions of The Incomplete Fibonacci And Lucas Numbers. Rendiconti Del Circolo Matematico Di Palermo, Xlviii, 91–596, 9.
- Ramírez, J. (2015). Incomplete generalized Fibonacci and Lucas polynomials. Hacettepe Journal of Mathematics and Statistics, 44(2), 363–373.
- Ramírez, J. (2013). Incomplete
*k*-Fibonacci and*k*-Lucas numbers. Chinese Journal of Mathematics, Article ID 107145. - Roman, S. (1984). The umbral calculus, Pure and Applied Mathematics, 111. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1984, x+193 pp. ISBN: 0-12-594380-6.
- Srivastava, H. M., & Manocha, H. L. A. (1984). Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York.
- Shannon, A. G., & Horadam, A. F. (1971). Generating functions for powers of third order recurrence sequences. Duke Mathematical Journal, 38(4), 791–794.
- Shannon, A. G., & Bernstein, L. (1973). The Jacobi–Perron algorithm and the algebra of recursive sequences. Bulletin of the Australian Mathematical Society, 8(2), 261–277.
- Shannon, A. G., & Horadam, A. F. (1999). Some Relationships among Vieta, Morgan-Voyce and Jacobsthal Polynomials, in Fredric T Howard (ed.), Applications of Fibonacci Numbers, Volume 8. Dordrecht: Kluwer, 307–323.
- Taşçı, D., & Yalçın, F. (2013). Vieta–Pell and Vieta–Pell–Lucas polynomials. Advances in Difference Equations, 2013, Article no. 224.
- Uygun, Ş., Karataş, H., & Aytar, H. (2020). Notes on Generalization of Vieta–Pell and Vieta– Pell–Lucas polynomials. International Journal of Mathematics Research, 12(1), 5–22.

## Related papers

## Cite this paper

Kuloğlu, B., Özkan, E., & Shannon, A. G. (2021). Incomplete generalized Vieta–Pell and Vieta–Pell–Lucas polynomials. *Notes on Number Theory and Discrete Mathematics*, 27(4), 245-256, DOI: 10.7546/nntdm.2021.27.4.245-256.