Abstract
Our aim is to solve a system of functional equations closely related to trigonometric functional equations. This allows us to express quaternion-valued multiplicative functions in terms of complex-valued multiplicative functions. As an application of our results, we give the continuous quaternion-valued solutions of a functional equation on \(({\mathbb {R}},\cdot )\).
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References
Aczél, J., Dhombres, J.: Functional Equations in Several Variables. Cambridge Uni versity Press, Cambridge (1989)
Ebanks, B., Stetkær, H.: d’Alembert’s other functional equation. Publ. Math. Debr. 87(3/4), 319–349 (2015)
Ebanks, B.: The cosine and sine addition and subtraction formulas on semigroups. Acta Math. Hung. 165, 337–354 (2021)
Kannappan, Pl.: Functional Equations and Inequalities with Applications. Springer, New York (2009)
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)
Stetkær, H.: On multiplicative maps. Semigroup Forum 63, 466–468 (2001)
Stetkær, H.: Functional Equations on Groups. World Scientific Publishing Company, Singapore (2013)
Vince, J.: Quaternions for Computer Graphics. Springer, London (2011)
Zhang, F.: Quaternions and matrices of quaternions. Linear Algebra Appl. 251, 21–57 (1997)
Acknowledgements
The authors would like to express their most sincere gratitude to the referee for a number of constructive comments which have led to an essential improvement of the paper.
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The work is prepared by the authors. I Driss Zeglami as director of doctoral thesis of Ayoub Ouhbi and Mohamed Ayoubi as co-supervisor. I declare that the elaboration of this work evolved in a natural way. All the authors contributed to formulating the question, writing, revising and reviewing the manuscript.
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Ouhabi, A., Zeglami, D. & Ayoubi, M. Quaternion-valued multiplicative functions on semigroups. Aequat. Math. (2024). https://doi.org/10.1007/s00010-024-01040-w
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DOI: https://doi.org/10.1007/s00010-024-01040-w