A generalization of d’Alembert’s other functional equation on semigroups

Abstract

Given a semigroup S generated by its squares and equipped with an involutive automorphism \(\sigma \) and a multiplicative function \(\mu :S\rightarrow \mathbb {C}\) such that \(\mu (x\sigma (x))=1\) for all \(x\in S\), we determine the complex-valued solutions of the following functional equation

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

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Correspondence to Omar Ajebbar.

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Ajebbar, O., Elqorachi, E. A generalization of d’Alembert’s other functional equation on semigroups. Aequat. Math. 94, 913–930 (2020). https://doi.org/10.1007/s00010-019-00694-1

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Keywords

  • Semigroup
  • Involutive automorphism
  • Multiplicative function
  • d’Alembert equation

Mathematics Subject Classification

  • Primary 39B52
  • Secondary 39B32