Iterative characterizations of powers and exponentials

Summary

Two theorems are proved concerning characterization of powers and exponentials by iterated (composite) functional equations in one complex variable for entire functions. It is shown that:

The only non-constant entire function satisfying the functional equation

$$F(z + F(z)) = F(z)F(z^{k - 1} + 1),$$

((1))

wherek is a fixed integer ⩾ 3, isF(z) = z ^k; and

The only non-constant entire functions satisfying the functional equation

$$F(z + F(z)) = F'(F(z))F'(z) = \frac{d}{{dz}}(F(F(z)))$$

((2))

are$\frac{1}{{a^2 }}e^{az}$ wherea is an arbitrary non-zero complex constant.