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: firstname.lastname@example.org)
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-analoguesReferences:
- 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.
- 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 q-Lucas polynomials, Appl. Math. Comput. 226, 691–698, 2014.
- 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.
- 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. q-Fibonacci polynomials, Fibonacci Quart. 13, 97–102, 1975.
- J. Cigler, q-Fibonacci polynomials, Fibonacci Quart. 41 (1), 31–40, 2003.
- J. Cigler, A new class of q-Fibonacci polynomials, Electron. J. Comb. 10 (1), 2003; Article ID: R19.
- J. Cigler, q-Fibonacci polynomials and the Rogers-Ramanujan identities, Ann. Comb. 8 (3), 269–285, 2004.
- J. Cigler, Some beautiful q-analogue of Fibonacci and Lucas polynomials, ArXiv:11042699, 2011.
- C. Kizilates, M. Çetin and N. Tuglu, q-Generalization of biperiodic Fibonacci and Lucas numbers, J. Math. Anal. 8 (5), 71–85, 2017.
- 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.
- S. Vajda, Fibonacci and Lucas numbers, and the golden section, theory and applications, Dover publication, INC. Mineola, New York, 2007.