Let \(\mathbb {K}\) be either \(\mathbb {R}\) or \(\mathbb {C}\) and \(\alpha \in \mathbb {R}\). We determine the solutions \(f:\mathbb {R}^{3}\rightarrow \mathbb { K}\) of the following new parametric functional equation:

$$\begin{aligned}&f(x_{1}x_{2}+\alpha y_{1}z_{2}+\alpha y_{2}z_{1},x_{1}y_{2}+x_{2}y_{1}+\alpha z_{1}z_{2},x_{1}z_{2}+x_{2}z_{1}+y_{1}y_{2}) \\&\quad =f(x_{1},y_{1},z_{1})f(x_{2},y_{2},z_{2}),\ (x_{1},y_{1},z_{1}),(x_{2},y_{2},z_{2})\in \mathbb {R}^{3}, \end{aligned}$$

which results from the product of two numbers in a cubic free field. We equip \(\mathbb {R}^{3}\) with a binary operation to show that the non-zero solutions of this equation are monoid homomorphisms and we investigate our results to introduce and find the solutions of d’Alembert’s functional equations with endomorphisms.

中文翻译：

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源自立方数域中的乘积的函数方程

让\(\mathbb {K}\)是\(\mathbb {R}\)或\(\mathbb {C}\)和\(\alpha \in \mathbb {R}\)。我们确定以下新参数函数方程的解\(f:\mathbb {R}^{3}\rightarrow \mathbb { K}\)：

$$\begin{对齐}&f(x_{1}x_{2}+\alpha y_{1}z_{2}+\alpha y_{2}z_{1},x_{1}y_{2}+x_ {2}y_{1}+\alpha z_{1}z_{2},x_{1}z_{2}+x_{2}z_{1}+y_{1}y_{2}) \\&\ quad =f(x_{1},y_{1},z_{1})f(x_{2},y_{2},z_{2}),\ (x_{1},y_{1},z_ {1}),(x_{2},y_{2},z_{2})\in \mathbb {R}^{3}, \end{aligned}$$

这是由三次自由场中的两个数字的乘积得出的。我们为\(\mathbb {R}^{3}\)配备了一个二元运算来证明这个方程的非零解是幺半群同态，我们研究我们的结果以引入和找到 d'Alembert 函数方程的解与内同态。