Several Determinantal Expressions of Generalized Tribonacci Polynomials and Sequences


  • Can Kızılateş Department of Mathematics, Faculty of Arts and Sciences, Zonguldak Bülent Ecevit University, Zonguldak, Turkey
  • Wei-Shih Du Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 82444, Taiwan
  • Feng Qi



Generalized Tribonacci sequence, Tribonacci polynomial, Tribonacci number, Hessenberg determinant


In the paper, the authors present several explicit formulas for the $(p,q,r)$-Tribonacci polynomials and generalized Tribonacci sequences in terms of the Hessenberg determinants and, consequently, derive several explicit formulas for the Tribonacci numbers and polynomials, the Tribonacci--Lucas numbers, the Perrin numbers, the Padovan (Cordonnier) numbers, the Van der Laan numbers, the Narayana numbers, the third order Jacobsthal numbers, and the third order Jacobsthal--Lucas numbers in terms of special Hessenberg determinants.


N. Bourbaki, Functions of a Real Variable, Elementary Theory, Translated from the 1976 French original by Philip Spain, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2004; available online at

N. D. Cahill, J. R. D’Errico, D. A. Narayan and J. Y. Narayan, Fibonacci determinants, College Math. J. 33 (2002), No. 3, 221–225; available online at

G. Cerda-Morales, On the third-order Jacobsthal and third-order Jacobsthal-Lucas sequences and their matrix representations, arXiv preprint (2018), available online at

G. Cerda-Morales, Some results on dual third-order Jacobsthal quaternions, Filomat 33 (2019), No. 7, 1865–1876; available online at

G. Cerda-Morales, Third-order Jacobsthal generalized quaternions, J. Geom. Sym- metry Phys. 50 (2018), 11–27; available online at

J. L. Cereceda, Determinantal representations for generalized Fibonacci and Tribonacci num- bers, Int. J. Contemp. Math. Sci. 9 (2014), No. 6, 269–285; available online at

C. K. Cook and M. R. Bacon, Some identities for Jacobsthal and Jacobsthal–Lucas numbers satisfying higher order recurrence relations, Ann. Math. Inform. 41 (2013), 27–39.

M. C. Dağlı and F. Qi, Several closed and determinantal forms for convolved Fibonacci num- bers, J. Appl. Math. Stat. Inform., 17(2021), No. 1; available online at

T. V. Didkivska and M. V. Stopochkina, Properties of Fibonacci–Narayana numbers, World Math. 9 (2003), No. 1, 29–36.

M. Elia, Derived sequences, the Tribonacci recurrence and cubic forms, Fibonacci Quart. 39 (2001), No. 2, 107–115.

M. Feinberg, Fibonacci-Tribonacci, Fibonacci Quart. 1(1963), No.3, 70–74.

V. E. Hoggatt and M. Bicknell, Generalized Fibonacci polynomials, Fibonacci Quart. 11 (1973), No. 5, 457–465.

S. Hu and M.-S. Kim, Two closed forms for the Apostol–Bernoulli polynomials, Ramanu- jan J. 46 (2018), No. 1, 103–117; available online at


C. Kızılateş, W.-S. Du, and F. Qi, Several determinantal expressions of generalized Tribonacci polynomials and sequences, Authorea Preprints (2020), available online at

T. Koshy, Fibonacci and Lucas Numbers with Applications, Vol. 1, Second edition, Pure and Applied Mathematics (Hoboken), John Wiley & Sons, Inc., Hoboken, NJ, 2018.

G. Y. Lee and M. Asci, Some properties of the (p, q)-Fibonacci and (p, q)-Lucas polynomials, J. Appl. Math. 2012. Article ID 264842; 18 pages; available online at

S. Pethe, Some identities for Tribonacci sequences, Fibonacci Quart. 26 (1988), No.2, 144–151.

F. Qi, A determinantal expression and a recursive relation of the Delannoy numbers, Acta Univ. Sapientiae Math. 13 (2021), No. 1, in press; available online at

F. Qi, Derivatives of tangent function and tangent numbers, Appl. Math. Comput. 268(2015), 844–858; available online at

F. Qi, Determinantal expressions and recurrence relations for Fubini and Eulerian polynomials, J. Interdiscip. Math. 22 (2019), No. 3, 317–335; available online at

F. Qi, V. Čerňanová and Y. S. Semenov, Some tridiagonal determinants related to central Delannoy numbers, the Chebyshev polynomials, and the Fibonacci polynomials, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 81 (2019), No. 1, 123–136.

F. Qi, M. C. Dağlı, and W.-S. Du, Determinantal forms and recursive relations of the Delan- noy two-functional sequence, Adv. Theory Nonlinear Anal. Appl. 4(2020), No. 3, 184–193; available online at

F. Qi and B.-N. Guo, A diagonal recurrence relation for the Stirling numbers of the first kind, Appl. Anal. Discrete Math. 12 (2018), No. 1, 153–165; available online at

F. Qi andB.-N. Guo, Some determinantal expressions and recurrence relations of the Bernoulli polynomials, Mathematics 4 (2016), No. 4, Article 65, 11 pages; available online at

F. Qi, C. Kızılateş, and W.-S. Du, A closed formula for the Horadam polynomials in terms of a tridiagonal determinant, Symmetry 11 (2019), No. 6, 8 pages; available online at

F. Qi, O. Kouba, and I. Kaddoura, Computation of several Hessenberg determinants, Math. Slovaca 70 (2020), No. 6, 1521–1537; available online at,

F. Qi, X.-T. Shi, and F.-F. Liu, Several identities involving the falling and rising factorials and the Cauchy, Lah, and Stirling numbers, Acta Univ. Sapientiae Math. 8 (2016), No. 2, 282–297; available online at

F. Qi and J.-L. Zhao, Some properties of the Bernoulli numbers of the second kind and their generating function, Bull. Korean Math. Soc. 55 (2018), No. 6, 1909–1920; available online at

F. Qi, J.-L. Zhao, and B.-N. Guo, Closed forms for derangement numbers in terms of the Hessenberg determinants, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RAC- SAM 112 (2018), No. 4, 933–944; available online at

A. G. Shannon, P. G. Anderson, and A. F. Horadam, Properties of Cordonnier, Perrin and Van der Laan numbers, Internat. J. Math. Ed. Sci. Tech. 37 (2006), No. 7, 825–831; available online at

A. G. Shannon and A. F. Horadam, Some properties of third-order recurrence relations, Fi- bonacci Quart. 10 (1972), No. 2, 135–145.

M. Shattuck and E. Tan, Incomplete generalized (p,q,r)-Tribonacci polynomials, Appl. Appl. Math. 13 (2018), No. 1, 1–18.

M. Tan and Y. Zhang, A note on bivariate and trivariate Fibonacci polynomials, Southeast Asian Bull. Math. 29 (2005), No. 5, 975–990.

C. C. Yalavigi, Properties of Tribonacci numbers, Fibonacci Quart.10(1972), No.3, 231–246.




How to Cite

Kızılateş, C., Du, W.-S., & Qi, F. (2022). Several Determinantal Expressions of Generalized Tribonacci Polynomials and Sequences. Tamkang Journal of Mathematics, 53(3), 277–291.




Most read articles by the same author(s)

1 2 > >>