$d$-Gaussian Fibonacci, $d$-Gaussian Lucas polynomials and their matrix representations

E.  Özkan; M.  Uysal

doi:10.37863/umzh.v75i4.6988

E. Özkan Department of Mathematics, Erzincan Binali Yıldırım University, Faculty of Arts and Sciences, Turkey
M. Uysal Graduate School of Natural and Applied Sciences, Erzincan Binali Yıldırım University, Turkey

DOI: https://doi.org/10.37863/umzh.v75i4.6988

Keywords: d-Gaussian Fibonacci polynomials,, d-Gaussian Lucas polynomials,, Generating function, Binet formula, Riordan matrix.

Abstract

UDC 517.5

We define $d$-Gaussian Fibonacci polynomials and $d$-Gaussian Lucas polynomials. We present the matrix representations of these polynomials. By using the Riordan method, we obtain the factorizations of the Pascal matrix including the polynomials. In addition, we define the infinite $d$-Gaussian Fibonacci polynomial matrix and the $d$-Gaussian Lucas polynomial matrix and give their inverses.

References

G. Berzsenyi, Gaussian Fibonacci numbers, Fibonacci Quart., 15, 223–236 (1977).

S. Falcón, Á. Plaza, On the Fibonacci $k$-numbers, Chaos, Solitons and Fractals, 32, № 5, 1615–1624 (2007); https://doi.org/10.1016/j.chaos.2006.09.022. DOI: https://doi.org/10.1016/j.chaos.2006.09.022

S. Falcon, Á. Plaza, The $k$-Fibonacci sequence and the Pascal 2-triangle, Chaos, Solitons and Fractals, 33, № 1, 38–49 (2007); https://doi.org/10.1016/j.chaos.2006.10.022. DOI: https://doi.org/10.1016/j.chaos.2006.10.022

V. E. Hoggatt (Jr.), M. Bicknell, Roots of Fibonacci polynomials, Fibonacci Quart., 11, № 3, 271–274 (1973).

J. H. Jordan, Gaussian Fibonacci and Lucas numbers, Fibonacci Quart., 3, № 4, 315–318 (1965).

T. Koshy, Fibonacci and Lucas numbers with applications, John Wiley & Sons (2019).

A. Nalli, P. Haukkanen, On generalized Fibonacci and Lucas polynomials, Chaos, Solitons and Fractals, 42, № 5, 3179–3186 (2009); https://doi.org/10.1016/j.chaos.2009.04.048. DOI: https://doi.org/10.1016/j.chaos.2009.04.048

E. Özkan, M. Taştan, On Gauss Fibonacci polynomials, on Gauss Lucas polynomials and their applications, Commun. Algebra, 48, № 3, 952–960 (2020); https://doi.org/10.1080/00927872.2019.1670193. DOI: https://doi.org/10.1080/00927872.2019.1670193

E. Özkan, I. Altun, Generalized Lucas polynomials and relationships between the Fibonacci polynomials and Lucas polynomials, Commun. Algebra, 47, 10–12 (2019); https://doi.org/10.1080/00927872.2019.1576186. DOI: https://doi.org/10.1080/00927872.2019.1576186

E. Özkan, M. Taştan, A new families of Gauss $k$-Jacobsthal numbers and Gauss $k$-Jacobsthal–Lucas numbers and their polynomials, J. Sci. and Arts, 4, 893–908 (2020). DOI: https://doi.org/10.46939/J.Sci.Arts-20.4-a10

E. Özkan, B. Kuloǧlu, On the new Narayana polynomials, the Gauss Narayana numbers and their polynomials, Asian-Eur. J. Math., 14, № 6, Article~2150100 (2021); https://doi.org/10.1142/S179355712150100X. DOI: https://doi.org/10.1142/S179355712150100X

E. Özkan, M. Taştan, On a new family of Gauss $k$-Lucas numbers and their polynomials, Asian-Eur. J. Math., 14, № 6, Article~2150101 (2021); https://doi.org/10.1142/S1793557121501011. DOI: https://doi.org/10.1142/S1793557121501011

B. Sadaoui, A. Krelifa, $d$-Fibonacci and $d$-Lucas polynomials, J. Math. Model., 9, № 3, 1–12 (2021); https://doi.org/ 10.22124/JMM.2021.17837.1538.

L. W. Shapiro, S. Getu, W. J. Woan, L. C. Woodson, The riordan group, Discrete Appl. Math., 34, № 1-3, 229–239 (1991). DOI: https://doi.org/10.1016/0166-218X(91)90088-E

M. Taştan, E. Özkan, A. G. Shannon, The generalized $k$-Fibonacci polynomials and generalized $k$-Lucas polynomials, Notes Number Theory and Discrete Math., 27, № 2, 148–158 (2021); https://doi.org/10.7546/ nntdm.2021.27.2.148-158. DOI: https://doi.org/10.7546/nntdm.2021.27.2.148-158