Skip to main content
SpringerLink
Log in

A Functional Equation Originated from the Product in a Cubic Number Field

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

Let \(\mathbb {K}\) be either \(\mathbb {R}\) or \(\mathbb {C}\) and \(\alpha \in \mathbb {R}\). We determine the solutions \(f:\mathbb {R}^{3}\rightarrow \mathbb { K}\) of the following new parametric functional equation:

$$\begin{aligned}&f(x_{1}x_{2}+\alpha y_{1}z_{2}+\alpha y_{2}z_{1},x_{1}y_{2}+x_{2}y_{1}+\alpha z_{1}z_{2},x_{1}z_{2}+x_{2}z_{1}+y_{1}y_{2}) \\&\quad =f(x_{1},y_{1},z_{1})f(x_{2},y_{2},z_{2}),\ (x_{1},y_{1},z_{1}),(x_{2},y_{2},z_{2})\in \mathbb {R}^{3}, \end{aligned}$$

which results from the product of two numbers in a cubic free field. We equip \(\mathbb {R}^{3}\) with a binary operation to show that the non-zero solutions of this equation are monoid homomorphisms and we investigate our results to introduce and find the solutions of d’Alembert’s functional equations with endomorphisms.

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. Aczel, J.: Lectures on Functional Equations and Their Applications. Academic Press, New York (1966)

    MATH  Google Scholar 

  2. Ayoubi, M., Zeglami, D.: D’Alembert’s functional equations on monoids with an anti-endomorphism. Results Math. 75, 74 (2020)

    Article  MathSciNet  Google Scholar 

  3. Berrone, L.R., Dieulefait, L.V.: A functional equation related to the product in a quadratic number field. Aequat. Math. 81(1–2), 167–175 (2011)

    Article  MathSciNet  Google Scholar 

  4. Chávez, E.A., Sahoo, P.K.: On functional equations of the multiplicative type. J. Math. Anal. Appl. 431, 283–299 (2015)

    Article  MathSciNet  Google Scholar 

  5. Chung, J.K., Sahoo, P.K.: General solution of some functional equations related to the determinant of symmetric matrices. Demonstr. Math. 35, 539–544 (2002)

    MathSciNet  MATH  Google Scholar 

  6. Ebanks, B.: Reconsidering a functional equation arising from number theory. Publ. Math. Debrecen 86, 135–147 (2015)

    Article  MathSciNet  Google Scholar 

  7. Fadli, B., Kabbaj, S., Sabour, Kh, Zeglami, D.: Functional equations on semigroups with an endomorphism. Acta Math. Hungar. 150, 363–371 (2016)

    Article  MathSciNet  Google Scholar 

  8. Houston, K.B., Sahoo, P.K.: On a functional equation related to the determinant of symmetric two-by-two matrices. Sarajevo J. Math. 3, 1–12 (2007)

    MathSciNet  MATH  Google Scholar 

  9. Houston, K.B., Sahoo, P.K.: On two functional equations and their solutions. Appl. Math. Lett. 21, 974–977 (2008)

    Article  MathSciNet  Google Scholar 

  10. Jung, S.-M., Bae, J.-H.: Some functional equations originating from number theory. Proc. Indian Acad. Sci. 113, 91–98 (2003)

    MathSciNet  MATH  Google Scholar 

  11. Mouzoun, A., Zeglami, D., Aissi, Y.: Matrix homomorphism equations on a class of monoids and non abelian groupoids. J. Math. Anal. Appl. 503, 125354 (2021)

  12. Sinopoulos, P.: Functional equations on semigroups. Aequat. Math. 59, 255–261 (2000)

    Article  MathSciNet  Google Scholar 

  13. Stetkæ, H.: Functional Equations on Groups. World Scientific Publishing Company, Singapore (2013)

    Book  Google Scholar 

  14. Stetkær, H.: A variant of d’Alembert’s functional equation. Aequat. Math. 89, 657–662 (2015)

    Article  MathSciNet  Google Scholar 

  15. Zeglami, D.: Some functional equations related to number theory. Acta Math. Hungar. 149, 490–508 (2016)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, E.N.S.A.M, Moulay ISMAÏL University, B.P: 15290, Al Mansour, Meknes, Morocco

    A. Mouzoun, D. Zeglami & M. Ayoubi

Authors
  1. A. Mouzoun
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. Zeglami
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Ayoubi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. Zeglami.

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

Mouzoun, A., Zeglami, D. & Ayoubi, M. A Functional Equation Originated from the Product in a Cubic Number Field. Mediterr. J. Math. 18, 191 (2021). https://doi.org/10.1007/s00009-021-01858-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-021-01858-7

Keywords

Mathematics Subject Classification

Search

Navigation