Skip to main content
SpringerLink
Log in

The Cosine–Sine functional equation on a semigroup with an involutive automorphism

  • Published:
Aequationes mathematicae Aims and scope Submit manuscript

Abstract

We determine the complex-valued solutions of the following extension of the Cosine–Sine functional equation

$$\begin{aligned} f(x\sigma (y))=f(x)g(y)+g(x)f(y)+h(x)h(y),\quad x,y\in S, \end{aligned}$$

where S is a semigroup generated by its squares and \(\sigma \) is an involutive automorphism of S. We express the solutions in terms of multiplicative and additive functions.

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. Aczél, J.: Lectures on functional equations and their applications. In: Aczél, J. (ed.) Mathematics in Sciences and Engineering, vol. 19. Academic Press, New York (1966)

  2. Chung, J.K., Kannappan, Pl, Ng, C.T.: A generalization of the Cosine–Sine functional equation on groups. Linear Algebra Appl. 66, 259–277 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ebanks, B., Stetkær, H.: d’Alembert’s other functional equation on monoids with involution. Aequ. Math. 89(1), 187–206 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Glaeser, G.: Dérivation des algèbres commutatives. C. R. Acad. Sci. Paris 233, 1550–1552 (1951)

    MathSciNet  MATH  Google Scholar 

  5. Poulsen, T.A., Stetkær, H.: On the trigonometric subtraction and addition formulas. Aequ. Math. 59(1–2), 84–92 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. Sahoo, P.K., Kannappan, P.: Introduction to Functional Equations. Chapman & Hall/CRC Press, Boca Raton (2011)

    MATH  Google Scholar 

  7. Stetkær, H.: Functional Equations on Groups. World Scientific Publishing Co, Singapore (2013)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Sciences, Ibn Zohr University, Agadir, Morocco

    Omar Ajebbar & Elhoucien Elqorachi

Authors
  1. Omar Ajebbar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Elhoucien Elqorachi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Omar Ajebbar.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ajebbar, O., Elqorachi, E. The Cosine–Sine functional equation on a semigroup with an involutive automorphism. Aequat. Math. 91, 1115–1146 (2017). https://doi.org/10.1007/s00010-017-0512-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00010-017-0512-9

Mathematics Subject Classification

Keywords

Search

Navigation