On the general solution of the system of cocycle equations without regularity conditions

Summary.

We describe the general solution (α, β), where $\alpha = (\alpha (s,x))_{s \in \mathbb{C}}$ and $\beta = (\beta (s,x))_{s \in \mathbb{C}}$ are families of formal power series in $\mathbb{C}\left[\kern-0.15em\left[ x \right]\kern-0.15em\right],$ of the two so-called cocycle equations

$$ \alpha (s + t,x) = \alpha (s,x)\alpha (t,\pi (s,x)),\quad s,t \in \mathbb{C} $$

((Co1))

$$ \beta (s + t,x) = \beta (s,x)\alpha (t,\pi (s,x)) + \beta (t,\pi (s,x)),\quad s,t \in \mathbb{C} $$

((Co2))

together with the boundary condition

$$ \alpha (0,x) = 1,\quad \beta (0,x) = 0, $$

((B1))

where $\pi = (\pi (s,x))_{s \in \mathbb{C}}$ is an iteration group in $\mathbb{C}\left[\kern-0.15em\left[ x \right]\kern-0.15em\right].$ Our method is based on the knowledge of the regular solutions of (Co1) and (Co2) and on a well-known and often used theorem concerning the algebraic relations between exponential functions and additive functions.