Addition formulae with singularities

Abstract

We deal with functional equations of the form f(x+y) = F(f(x)J(y)) (so called addition formulas) assuming that the given binary operation F is associative but its domain of definition is disconnected (admits "singularities"). The function Flu.v) := serves here as a good example; the corresponding equation characterizes the hyperpolic tangent. Our considerations may be viewed as counterparts of L. Losonczi's [4] and K. Domańska's [2] results on local solutions of the functional equation f(F(x,y)) = f(x) + f(y) with the same behaviour of the given associative operation F. Our results exhibit a crucial role of 1 that turns out to be the critical value towards the range of the unknown function. What concerns the domain we admit fairly general structures (groupoids, groups, 2-divisible groups). In the case where the domain forms a group admitting subgroups of index 2 the family of solutions enlarges considerably.