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Some functional equations related to number theory

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Abstract

We introduce a new functional equation (E(α)), \({\alpha \geqq 0}\) which is originating from the product in the number field \({\mathbb{Q}\left(\sqrt[4]{\alpha}\,\right)}\). We give an explicit description of the solutions \({f : \mathbb{R}^{4}\to \mathbb{R}}\) of this equation for \({\alpha \geqq 0}\) and investigate these results to find the solutions \({f : \mathbb{R}^{4} \to \mathbb{C}}\) of d’Alembert’s type and a Van Vleck’s functional equations originating from number theory. Our considerations refer to the paper [2] in which L. R. Berrone and L. Dieulefait determine, for a fixed real \({\alpha}\), the real valued solutions of the equation

$$ f(x_{1},y_{1})f(x_{2},y_{2})=f(x_{1}x_{2}+\alpha y_{1}y_{2},x_{1}y_{2}+x_{2}y_{1}),\quad (x_{1},y_{1}),(x_{2},y_{2})\in \mathbb{R}^{2}.$$

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

    D. Zeglami

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  1. D. Zeglami
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Zeglami, D. Some functional equations related to number theory. Acta Math. Hungar. 149, 490–508 (2016). https://doi.org/10.1007/s10474-016-0634-x

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  • DOI: https://doi.org/10.1007/s10474-016-0634-x

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