International Journal of Mathematics and Mathematical Sciences
Volume 5 (1982), Issue 2, Pages 315-335

The solution of the functional equation of D'Alembert's type for commutative groups

A. L. Rukhin

Department of Statistics, Purdue University, West Lafayette 47907, Indiana, USA

Received 2 June 1980; Revised 11 December 1980

Copyright © 1982 A. L. Rukhin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A functional equation of the form ϕ1(x+y)+ϕ2(xy)=inαi(x)βi(y), where functions ϕ1,ϕ2,αi,βi, i=1,,n are defined on a commutative group, is solved. We also obtain conditions for the solutions of this equation to be matrix elements of a finite dimensional representation of the group.