**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}

^{a}USTHB, Faculty of Mathematics, RECITS Lab., Po. Box 32, El Alia 16111, Bab Ezzouar, Algiers, Algeria

^{b}Fac. 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

^{c}Depart. 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: hbelbachir@usthb.dz)

**Full Text:** PDF

**Abstract**

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

**References:**

- S. Abbad, H. Belbachir and B. Benzaghou,
*Companion sequences associated to the*, Turkish J. Math.*r*-Fibonacci sequence: algebraic and combinatorial properties**43 (3)**, 1095–1114, 2019. - S. Amrouche and H. Belbachir,
*Unimodality and linear recurrences associated with rays in the Delannoy triangle*, Turkish J. Math.**44 (1)**, 118–130, 2020. - 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. - H. Belbachir and F. Bencherif,
*On some properties of Chebyshev polynomials*, Discuss. Math. – Gen. Algebra Appl.**28**, 121–133, 2008. - H. Belbachir and F. Bencherif,
*Unimodality of sequences associated to Pell numbers*, Ars Combin.**102**, 305–311, 2011. - 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. - H. Belbachir and A. Benmezai,
*An alternative approach to Cigler’s*, Appl. Math. Comput.*q*-Lucas polynomials**226**, 691–698, 2014. - H. Belbachir and A. Benmezai,
*A*, C. R. Math. Acad. Sci. Paris*q*-analogue for bi^{s}nomial coefficients and generalized Fibonacci sequences**352 (3)**, 167–171, 2014. - 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. - L. Carlitz,
*Fibonacci notes. IV.*, Fibonacci Quart.*q*-Fibonacci polynomials**13**, 97–102, 1975. - J. Cigler,
, Fibonacci Quart.*q*-Fibonacci polynomials**41 (1)**, 31–40, 2003. - J. Cigler,
*A new class of*, Electron. J. Comb.*q*-Fibonacci polynomials**10 (1)**, 2003; Article ID: R19. - J. Cigler,
, Ann. Comb.*q*-Fibonacci polynomials and the Rogers-Ramanujan identities**8 (3)**, 269–285, 2004. - J. Cigler,
*Some beautiful*, ArXiv:11042699, 2011.*q*-analogue of Fibonacci and Lucas polynomials - C. Kizilates, M. Çetin and N. Tuglu,
, J. Math. Anal.*q*-Generalization of biperiodic Fibonacci and Lucas numbers**8 (5)**, 71–85, 2017. - J. L. Ramirez and V. Sirvent,
*A*, J. Integer Seq.*q*-analogue of the biperiodic Fibonacci sequences**19 (2)**, 2016; Article ID: 16.4.6. - S. Vajda,
*Fibonacci and Lucas numbers, and the golden section, theory and applications*, Dover publication, INC. Mineola, New York, 2007.