Article ID: MTJPAM-D-21-00071

Title: The q-analogue of a specific property of second order linear recurrences

Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-21-00071; Volume 5 / Issue 3 / Year 2023, Pages 49-57

Document Type: Research Paper

Author(s): Hacène Belbachir a, Athmane Benmezai b, Abdelkader Bouyakoub c

aUSTHB, Faculty of Mathematics, RECITS Lab., Po. Box 32, El Alia 16111, Bab Ezzouar, Algiers, Algeria

bFac. of Eco. & Manag. Sc., Univ. of Dely Brahim, RECITS Lab., Rue Ahmed Ouaked, Dely Brahim, Algiers, Algeria-USTHB–Faculty of Mathematics, RECITS Lab., Po. Box 32, El Alia 16111, Bab Ezzouar, Algiers, Algeria

cDepart. of Math., Fac. of Sc., Oran Univ., GEAN Lab., Po. Box 1524, ELM_Naouer, 31000, Oran, Algeria

Received:28 November 2021, Accepted:12 March 2022, Published:2 July 2022.

Corresponding Author: Hacène Belbachir (Email address:

Full Text: PDF


A translated recurrent sequence of rank two is related to the Fibonacci sequence, this property is generalized in this paper using a q-analogue of Fibonacci sequence suggested by J. Cigler. We give some specialization to the generalized Fibonacci and Lucas sequences.

Keywords: Fibonacci polynomials, Lucas polynomials, q-analogues

  1. S. Abbad, H. Belbachir and B. Benzaghou, Companion sequences associated to the r-Fibonacci sequence: algebraic and combinatorial properties, Turkish J. Math. 43 (3), 1095–1114, 2019.
  2. S. Amrouche and H. Belbachir, Unimodality and linear recurrences associated with rays in the Delannoy triangle, Turkish J. Math. 44 (1), 118–130, 2020.
  3. S. Amrouche, H. Belbachir and J. L. Ramírez, Unimodality, linear recurrences and combinatorial properties associated to rays in the generalized Delannoy matrix, J. Differ. Equ. Appl. 25 (8),1200–1215, 2019.
  4. H. Belbachir and F. Bencherif, On some properties of Chebyshev polynomials, Discuss. Math. – Gen. Algebra Appl. 28, 121–133, 2008.
  5. H. Belbachir and F. Bencherif, Unimodality of sequences associated to Pell numbers, Ars Combin. 102, 305–311, 2011.
  6. H. Belbachir and A. Benmezai, Expansion of Fibonacci and Lucas polynomials: An answer to Prodinger’s question, J. Integer Seq. 15 (2), 2012; Article ID: 12.7.6.
  7. H. Belbachir and A. Benmezai, An alternative approach to Cigler’s q-Lucas polynomials, Appl. Math. Comput. 226, 691–698, 2014.
  8. H. Belbachir and A. Benmezai, A q-analogue for bisnomial coefficients and generalized Fibonacci sequences, C. R. Math. Acad. Sci. Paris 352 (3), 167–171, 2014.
  9. H. Belbachir, T. Komatsu and L. Szalay, Linear recurrences associated to rays in Pascal’s triangle and combinatorial identities, Math. Slovaca 64 (2), 287–300, 2014.
  10. L. Carlitz, Fibonacci notes. IV. q-Fibonacci polynomials, Fibonacci Quart. 13, 97–102, 1975.
  11. J. Cigler, q-Fibonacci polynomials, Fibonacci Quart. 41 (1), 31–40, 2003.
  12. J. Cigler, A new class of q-Fibonacci polynomials, Electron. J. Comb. 10 (1), 2003; Article ID: R19.
  13. J. Cigler, q-Fibonacci polynomials and the Rogers-Ramanujan identities, Ann. Comb. 8 (3), 269–285, 2004.
  14. J. Cigler, Some beautiful q-analogue of Fibonacci and Lucas polynomials, ArXiv:11042699, 2011.
  15. C. Kizilates, M. Çetin and N. Tuglu, q-Generalization of biperiodic Fibonacci and Lucas numbers, J. Math. Anal. 8 (5), 71–85, 2017.
  16. J. L. Ramirez and V. Sirvent, A q-analogue of the biperiodic Fibonacci sequences, J. Integer Seq. 19 (2), 2016; Article ID: 16.4.6.
  17. S. Vajda, Fibonacci and Lucas numbers, and the golden section, theory and applications, Dover publication, INC. Mineola, New York, 2007.