A parametric functional equation originating from number theory


Let S be a semigroup and α,β∈ℝ. The purpose of this paper is to determine the general solution f:ℝ2→S of the following parametric functional equation
f(x1x2+αy1y2,x1y2+x2y1+βy1y2) = f(x1,y1)f(x2,y2),
for all (x1,y1), (x2,y2)∈ℝ2, that generalizes some functional equations arising from number theory and is connected with the characterizations of the determinant of matrices.


functional equation; number theory; character; multiplicative function; monoid; additive function

1. J. Aczél and J. Dhombres, Functional Equations in Several Variables, Encyclopedia of Mathematics and its Applications, 31, Cambridge University Press, Cambridge, 1989.
2. Y. Aissi and D. Zeglami, D’Alembert’s matrix functional equation with an endomorphism on abelian groups, Results Math. 75 (2020), no. 4, Paper No. 137, 17 pp.
3. Y. Aissi, D. Zeglami, and M. Ayoubi, A variant of d’Alemberts matrix functional equation, Ann. Math. Sil. 35 (2021), no. 1, 21–43.
4. M. Akkouchi and M.H. Lalaoui Rhali, General solutions of some functional equations, Bol. Mat. 12 (2005), no 1, 57–62.
5. L.R. Berrone and L.V. Dieulefait, A functional equation related to the product in a quadratic number field, Aequationes Math. 81 (2011), no. 1–2, 167–175.
6. E.A. Chávez and P.K. Sahoo, On functional equations of the multiplicative type, J. Math. Anal. Appl. 431 (2015), no. 1, 283–299.
7. J. Chung and P.K. Sahoo, On a functional equation arising from Proth identity, Commun. Korean Math. Soc. 31 (2016), no. 1, 131–138.
8. B. Ebanks, Reconsidering a functional equation arising from number theory, Publ. Math. Debrecen, 86 (2015), no. 1-2, 135–147.
9. S.-M. Jung and J.-H. Bae, Some functional equations originating from number theory, Proc. Indian Acad. Sci. Math. Sci. 113 (2003), no. 2, 91–98.
10. A. Mouzoun, D. Zeglami, and Y. Aissi, Matrix homomorphism equations on a class of monoids and non Abelian groupoids, J. Math. Anal. Appl. 503 (2021), no. 2, Paper No. 125354, 24 pp.
11. A. Mouzoun, D. Zeglami, and M. Ayoubi, A functional equation originated from the product in a cubic number field, Mediterr. J. Math. 18 (2021), no. 5, Paper No. 191, 14 pp.
12. P. Sinopoulos, Wilson’s functional equation for vector and matrix functions, Proc. Amer. Math. Soc. 125 (1997), no. 4, 1089–1094.
13. H. Stetkær, Functional Equations on Groups, World Scientific Publishing Company, Singapore, 2013.
14. H. Stetkær, D’Alembert’s and Wilson’s functional equations for vector and 22 matrix valued functions, Math. Scand. 87 (2000), no. 1, 115–132.
15. D. Zeglami, Some functional equations related to number theory, Acta Math. Hungar. 149 (2016), no. 2, 490–508.

Published : 2022-01-17

MouzounA., ZeglamiD., & AissiY. (2022). A parametric functional equation originating from number theory. Annales Mathematicae Silesianae, 36(1), 71-91. Retrieved from https://www.journals.us.edu.pl/index.php/AMSIL/article/view/13462

Aziz Mouzoun  mouzounposte@gmail.com
Department of Mathematics, E.N.S.A.M, Moulay ISMAÏL University, Morocco  Morocco
Driss Zeglami 
Department of Mathematics, E.N.S.A.M, Moulay ISMAÏL University, Morocco  Morocco
Youssef Aissi 
Department of Mathematics, E.N.S.A.M, Moulay ISMAÏL University, Morocco  Morocco

The Copyright Holders of the submitted text are the Author and the Journal. The Reader is granted the right to use the pdf documents under the provisions of the Creative Commons 4.0 International License: Attribution (CC BY). The user can copy and redistribute the material in any medium or format and remix, transform, and build upon the material for any purpose.

  1. License
    This journal provides immediate open access to its content under the Creative Commons BY 4.0 license (http://creativecommons.org/licenses/by/4.0/). Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license.
  2. Author’s Warranties
    The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s.
  3. User Rights
    Under the Creative Commons Attribution license, the users are free to share (copy, distribute and transmit the contribution) and adapt (remix, transform, and build upon the material) the article for any purpose, provided they attribute the contribution in the manner specified by the author or licensor.
  4. Co-Authorship
    If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.