On polynomial functions on non-commutative groups

Let G be a topological group. We investigate relations between two classes of “polynomial like” continuous functions on G defined, respectively, by the conditions 1) Δhn+1f=0 for every h∈G, and 2) Δhn+1Δhn⋯Δh1f=0 for every h1,⋯,hn+1∈G. It is shown that for many (but not all) groups these classes coincide. We consider also Montel type versions of the above conditions - when 1) and 2) hold only for h in a generating subset of G. Our approach is based on the study of the counterparts of the discussed classes for general representations of groups (instead of the regular representation).