We denote by $\mathcal{E}$ the set of functions $f : \mathbb{C} \rightarrow \mathbb{C}$ expandable as a power series with radius of convergence infinity. Using the functions $Q_n \in \mathcal{E}$ satisfying $Q_n(z+1) - Q_n(z) = nz^{n-1}$ for all $n \in \mathbb{N}^*$ and $z \in \mathbb{C}$, and the bound $|Q_n(z)| \leqslant a\,\mathrm{e}^{bn|z|}$ for constants $a,b \in \mathbb{R}_+^*$, deduce the existence of a solution in $\mathcal{E}$ to the equation $(E_h)$: $$\forall z \in \mathbb{C},\, f(z+1) - f(z) = h(z)$$ when $h \in \mathcal{E}$.
We denote by $\mathcal{E}$ the set of functions $f : \mathbb{C} \rightarrow \mathbb{C}$ expandable as a power series with radius of convergence infinity. Using the functions $Q_n \in \mathcal{E}$ satisfying $Q_n(z+1) - Q_n(z) = nz^{n-1}$ for all $n \in \mathbb{N}^*$ and $z \in \mathbb{C}$, and the bound $|Q_n(z)| \leqslant a\,\mathrm{e}^{bn|z|}$ for constants $a,b \in \mathbb{R}_+^*$, deduce the existence of a solution in $\mathcal{E}$ to the equation $(E_h)$:
$$\forall z \in \mathbb{C},\, f(z+1) - f(z) = h(z)$$
when $h \in \mathcal{E}$.