D'Alembert's equation and spherical functions (Q1339672)

The author gives a natural way of extending d'Alembert's and Wilson's functional equations to groups, starting from the observation that the solutions of d'Alembert's functional equation are \(Z_ 2\)-spherical functions an \(\mathbb{R}^ 2\). Using the basic theory for spherical functions the author deduces that their general solutions are joint eigenfunctions for a system of partial differential operators and finds a formula for the bounded solutions. Interesting examples are also given in which the set of solutions of d'Alembert's or Wilson's functional equation are found.