Some functional equations related to number theory

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• 作者：D. Zeglami
• 刊名：Acta Mathematica Hungarica
• 出版年：2016
• 出版时间：August 2016
• 年：2016
• 卷：149
• 期：2
• 页码：490-508
• 全文大小：789 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Sciences
  Mathematics
  
• 出版者：Akad茅miai Kiad贸, co-published with Springer Science+Business Media B.V., Formerly Kluwer Academic
• ISSN：1588-2632
• 卷排序：149

文摘

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