Skip to main content
SpringerLink
Log in

On the Hankel matrices of finite rank and generalized Fibonacci sequences

  • Original Article
  • Published:
Boletín de la Sociedad Matemática Mexicana Aims and scope Submit manuscript

Abstract

In this paper, we propose an approach that utilizes the generalized Fibonacci sequence as a tool for studying the infinite Hankel matrix of finite rank, associated with a nonzero sequence \(S = \{s_n\}_ {n\ge 0}\) of real numbers. We offer detailed characterizations of both the generating and representing measures for any sequence associated with a Hankel matrix of finite rank. Furthermore, we make use of established results on the Z-transform of generalized Fibonacci sequences to investigate the connection between infinite Hankel matrices of finite rank and the Hamburger moment problem for such sequences. Indeed, we present some rephrased findings and newly derived results. Additionally, illustrative examples are provided to demonstrate the efficacy of our approach.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bakan, A., Berg, C.: Solvability of the Hankel Determinant Problem for Real Sequences. Contemporary Mathematics and Its Applications: Monographs, Expositions and Lecture Notes Frontiers in Orthogonal Polynomials and q-Series, World Scientific, pp. 85–117 (2018)

  2. Gantmacher, F.R.: The Theory of Matrices, vol. 2. AMS Chelsea Publishing, Providence (2000)

    Google Scholar 

  3. Berg, C., Szwarc, R.: A determinant characterization of moment sequences with finitely many mass-points. Linear Multilinear Algebra 1568–1576 (2015). https://doi.org/10.1080/03081087.2014.954516

  4. Iohvidov, I.S.: Hankel and Toeplitz Matrices and Forms: Algebraic Theory. Birkhäuser, Boston (1982)

    Google Scholar 

  5. Hasan, M.A., Hasan, A.A.: Hankel matrices of finite rank with applications to signal processing and polynomials. J. Math. Anal. Appl. 208, 218–242 (1997)

    Article  MathSciNet  Google Scholar 

  6. Curto, R.E., Fialkow, L.A.: Recursiveness, positivity, and truncated moment problems. Houst. J. Math. 17, 603–635 (1991)

    MathSciNet  Google Scholar 

  7. Curto, R.E., Fialkow, L.A.: Solution of the truncated complex moment problem for flat data. Mem. Am. Math. Soc. 119(568), x+52 (1996)

  8. Curto, R.E., Fialkow, L.A.: Flat extensions of positive moment matrices: recursively generated relations. Mem. Am. Math. Soc. 136(648), x+56 (1998)

  9. Al’pin, Yu.A.: The Hankel matrix rank theorem revised. Linear Algebra Appl. 537, 97–101 (2017)

    Article  MathSciNet  Google Scholar 

  10. Ben Taher, R., Rachidi, M.: Solving some generalized Vandermonde systems and inverse of their associate matrices via new approaches for the Binet formula. Appl. Math. Comput. 290, 267–280 (2016)

    MathSciNet  Google Scholar 

  11. Ben Taher, R., Rachidi, M., Zerouali, E.H.: Recursive subnormal completion and the truncated moment problem. Bull. Lond. Math. Soc. 33, 425–432 (2001)

    Article  MathSciNet  Google Scholar 

  12. Bernoussi, B., Rachidi, M., Saeki, O.: Factorial Binet formula and distributional moment formulation of generalized Fibonacci sequences. Soc. Ind. Appl. Math. Fibonacci Q. 42, 320–329 (2004)

    MathSciNet  Google Scholar 

  13. Kronecker, L.: ’Zur Theorie der Elimination einer Variabeln aus zwei algebraischen Gleichungen’. Monatsberichte der Koniglichen Preussische Akademie des Wissenschaften zu Berlin, Sitzung der phys.-math. Klasse vom 16, Juni 1881, 535–600 (1881)

  14. Kelly, W.G., Peterson, A.C.: Difference Equations: An Introduction with Applications. Academic Press, San Diego (1991)

    Google Scholar 

  15. Ben Taher, R., Rachidi, M., El Fetnassi, M.: On the z-transform and the non homogeneous linear difference equations. J. Interdiscip. Math. 7(3), 297–313 (2004)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are deeply grateful to the referees for their invaluable insights and constructive suggestions, which greatly improved the quality and presentation of the paper.

Author information

Author notes

  1. A. Taia and M. Lassri contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences, University Ibn Tofail, Kenitra, Morocco

    A. Taia

  2. Department of Mathematics, Faculty of Sciences, University Moulay Ismail, Meknes, Morocco

    M. Lassri & R. Ben Taher

Authors
  1. A. Taia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Lassri
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. R. Ben Taher
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. Ben Taher.

Ethics declarations

Conflict of interest

This study has not received any research funding. The authors declare that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Taia, A., Lassri, M. & Ben Taher, R. On the Hankel matrices of finite rank and generalized Fibonacci sequences. Bol. Soc. Mat. Mex. 30, 36 (2024). https://doi.org/10.1007/s40590-024-00614-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40590-024-00614-7

Keywords

Mathematics Subject Classification

Search

Navigation