Contribution to the study of two functional equations

Abstract

We improve a result of Mario Bonk on the functional equation¶¶$ f_1(u+v)f_2(u-v) = \sum_{i=1}^{k} g_i(u) h_i(v) $¶and we determine the general solution of the functional equation¶¶$ f(x+x',y+y',t+t'+ {xy'-yx' \over 2}) + f (x+y',y+x',t-t'+ {xx'-yy'\over 2}) =2f(x,y,t)f(x',y',t') $¶which was introduced by Henrik Stetkær as the defining identity for a K-spherical function on the Heisenberg group H, where K is a certain two-element subgroup of Aut(H).