d'Alembert's and Wilson's equations on Lie groups (Q1888648)

Let \(G\) be a connected, nilpotent Lie group. The author considers the functional equation \[ g(xy)+g(yx)+g(xy^{-1}) +g(y^{-1}x)=4g (x) g(y), \quad x,y\in G, \] which is a generalization of d'Alembert's functional equation. He shows that any solution \(g:G\to\mathbb{C}\), \(g\neq 0\), of it has the same form as on abelian groups, i.e., \(g(x)=[m(x)+m(x^{-1})]/2\), where \(m:g\to(\mathbb{C}\{0\},\cdot)\) is a homomorphism. He shows that the solutions of Wilson's functional equation \[ f(xy)+f(xy^{-1})= 2f(x)g (y),\quad x,y\in G, \] also have the same form as on abelian groups, except possibly in the case of \(g=1\) where the equation reduces to Jensen's functional equation. By solving Jensen's functional equation on the Heisenberg group he finds that it has a solution of another form than the one found for abelian groups. I. Corovei, C. T. Ng and the reviewer have presented a number of papers about the same functional equations on other types of non-abelian groups.