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A q-analogue of the bi-periodic Fibonacci and Lucas sequences and Rogers–Ramanujan Identities

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Abstract

In this paper, we define a Cigler-type q-analogue of the bi-periodic Fibonacci and Lucas polynomials. Also, we introduce two types of the q-analogue of the bi-periodic Lucas polynomials. We establish recurrence relations and several properties of these polynomials. Moreover, we obtain some related Rogers–Ramanujan type identities using the q-bi-periodic Lucas polynomials that allow an approach for Bailey pairs.

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References

  1. Andrews, G.E.: The Theory of Partitions. Addison-Wesley, Reading, MA, (1976); reissued: Cambridge University Press, Cambridge (1998)

  2. Andrews, G.E.: Fibonacci numbers and the Rogers–Ramanujan identities. Fibonacci Quart. 42, 3–19 (2004)

    MathSciNet  MATH  Google Scholar 

  3. Andrews, G.E., Berndt, B.C.: Ramanujan’s Lost Notebook Part I. Springer, New York (2005)

    Book  MATH  Google Scholar 

  4. Andrews, G.E., Berndt, B.C.: Ramanujan’s Lost Notebook Part II. Springer, New York (2009)

    MATH  Google Scholar 

  5. Andrews, G.E., Warnaar, S.O.: The Bailey transform and false theta functions. Ramanujan J. 14(1), 173–188 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bailey, W.N.: Some identities in combinatory analysis. Proc. Lond. Math. Soc. 2, 421–435 (1947)

    MathSciNet  MATH  Google Scholar 

  7. Bala, A., Verma, V.: Some properties of bi-variate bi-periodic Lucas polynomials. Ann. Romanian Soc. Cell Biol. 25, 8778–8784 (2021)

    Google Scholar 

  8. Belbachir, H., Benmezai, A.: An alternative approach to Cigler’s \(q\)-Lucas polynomials. Appl. Math. Comput. 226, 691–698 (2014)

    MathSciNet  MATH  Google Scholar 

  9. Belbachir, H., Benmezai, A.: A \(q\)-analogue for bi\(^s\)nomial coefficients and generalized Fibonacci sequences. C. R. Acad. Sci. Paris I(352), 167–171 (2014)

    Article  MATH  Google Scholar 

  10. Bilgici, G.: Two generalizations of Lucas sequence. Appl. Math. Comput. 245, 526–538 (2014). https://doi.org/10.1016/j.amc.2014.07

    Article  MathSciNet  MATH  Google Scholar 

  11. Carlitz, L.: Fibonacci notes. III. \(q\)-Fibonacci numbers. Fibonacci Quart. 12, 317–322 (1974)

    MathSciNet  MATH  Google Scholar 

  12. Carlitz, L.: Fibonacci notes. \(IV\). \(q\)-Fibonacci polynomials. Fibonacci Quart. 13, 97–102 (1975)

    MathSciNet  MATH  Google Scholar 

  13. Chu, W., Zhang, W.: Bilateral Bailey lemma and Rogers–Ramanujan identities. Adv. Appl. Math. 42, 358–391 (2009). https://doi.org/10.1016/j.aam.2008.07.003

    Article  MathSciNet  MATH  Google Scholar 

  14. Cigler, J.: A new class of \(q\)-Fibonacci polynomials. Electron. J. Combin. 10(1), 1–15 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cigler, J.: \(q\)-Fibonacci polynomials. Fibonacci Quart. 41(1), 31–40 (2003)

    MathSciNet  MATH  Google Scholar 

  16. Cigler, J.: \(q\)-Fibonacci polynomials and the Rogers–Ramanujan identities. Ann. Comb. 8(3), 269–285 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Cigler, J.: \(q\)-Lucas polynomials and associated Rogers–Ramanujan type identities. arXiv:0907.0165 (2009)

  18. Cigler, J.: Some beautiful \(q\)-analogues of Fibonacci and Lucas polynomials. arXiv:1104.2699 (2011)

  19. Edson, M., Yayenie, O.: A new generalization of Fibonacci sequences and extended Binet’s formula. Integers 9(A48), 639–654 (2009)

    MathSciNet  MATH  Google Scholar 

  20. Ekhad, S.B., Zeilberger, D.: The number of solutions of \(X^2=0\) in triangular matrices over \(GF(q)\). Electron. J. Combin. R2–R2 (1996)

  21. Koshy, T.: Fibonacci and Lucas numbers with applications. Wiley, New York (2001)

    Book  MATH  Google Scholar 

  22. Ramirez, J.L., Sirvent, V.: A \(q\)-analoque of the bi-periodic Fibonacci sequence. J. Integer Seq. 19 (2016)

  23. Rogers, L.J.: On two theorems of combinatory analysis and some allied identities. Proc. Lond. Math. Soc. 2(1), 315–336 (1917)

    Article  MATH  Google Scholar 

  24. Rowell, M.J.: The Bailey transform and conjugate Bailey pairs. The Pennsylvania State University (2007)

  25. Rowell, M.J.: A new general conjugate Bailey pair. Pacific J. Math. 238(2), 367–385 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Schur, I.: Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbrüche. S.-B. Preuss. Akad. Wiss. Phys. Math. Klasse, 302–321 (1917)

  27. Sills, A.V.: Finite Rogers–Ramanujan type identities. Electron. J. Combin. 10(1) R13, 1–122 (2003)

  28. Sills, A.V.: An invitation to the Rogers–Ramanujan Identities (2020)

  29. Tan, E.: A \(q\)-analogue of the bi-periodic Lucas sequence. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat 67(2), 220–228 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  30. Tan, E., Ekin, A.B.: Bi-periodic incomplete Lucas sequences. ARS Combin. 123, 371–380 (2015)

    MathSciNet  MATH  Google Scholar 

  31. Verma, A.B., Bala, A.: On properties of generalized bi-variate pi-periodic Fibonacci polynomials. Int. J. Adv. Sci. Technol. 29(3), 8065–8072 (2020)

    Google Scholar 

  32. Warnaar, S.O.: q-Hypergeometric proofs of polynomial analogues of the triple product identity, Lebesgue’s identity and Euler’s pentagonal number theorem. arXiv:math/0203229 (2002)

  33. Yayenie, O.: A note on generalized Fibonacci sequence. Appl. Math. Comput. 217(12), 5603–5611 (2011). https://doi.org/10.1016/j.amc.2010.12.038

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous reviewer for carefully reading our paper and offering corrections and helpful suggestions.

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Authors and Affiliations

  1. USTHB, Faculty of Mathematics, RECITS Laboratory, Po. Box 32, El Alia 16111, Bab Ezzouar, Algiers, Algeria

    Nassima Belaggoun, Hacène Belbachir & Athmane Benmezai

  2. CERIST: Research Center on Scientific and Technical Information 05, Rue des 3 frères Aissou, Ben Aknoun, Algiers, Algeria

    Nassima Belaggoun & Hacène Belbachir

  3. University of Dely Brahim, Faculty of Economy and Management Science, Rue Ahmed Ouaked, Dely Brahim, Algiers, Algeria

    Athmane Benmezai

Authors
  1. Nassima Belaggoun
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  2. Hacène Belbachir
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  3. Athmane Benmezai
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Correspondence to Nassima Belaggoun.

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This paper is partially supported by DGRSDT grant \(n^{\circ }\) C0656701.

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Belaggoun, N., Belbachir, H. & Benmezai, A. A q-analogue of the bi-periodic Fibonacci and Lucas sequences and Rogers–Ramanujan Identities. Ramanujan J 60, 693–728 (2023). https://doi.org/10.1007/s11139-022-00647-4

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  • DOI: https://doi.org/10.1007/s11139-022-00647-4

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