Some functional equations related to number theory (Q1677522)

Let \(\mathbb{R}\) denote the set of all real numbers and let \(0\leq\alpha\in\mathbb{R}.\) Define the operation \(\circ\) on \(\mathbb{R}^{4}\) by \[ (x_{1},y_{1},z_{1},w_{1})\circ (x_{2},y_{2},z_{2},w_{2})= \] \[ (x_{1}x_{2}+\alpha y_{1}w_{2}+\alpha w_{1}y_{2}+\alpha w_{1}w_{2}, x_{1}y_{2}+y_{1}x_{2}+\alpha z_{1}w_{2}+\alpha z_{2}w_{1}, \] \[ x_{1}z_{2}+x_{2}z_{1}+y_{1}y_{2}+\alpha w_{1}w_{2}, x_{1}w_{2}+x_{2}w_{1}+y_{1}z_{2}+z_{1}y_{2}). \] \smallskip In this paper, the author gives an explicite description of the solutions \(f:\mathbb{R}^{4}\to \mathbb{R}\) of the Cauchy functional equation \[ f((x_{1},y_{1},z_{1},w_{1})\circ (x_{2},y_{2},z_{2},w_{2}))=f(x_{1},y_{1},z_{1},w_{1})f(x_{2},y_{2},z_{2},w_{2}) \] and applies the result to find the solutions of d'Alembert types and a Van Vlek's functional equation originating from number theory.