A function $E$ (with rational coefficients) is a power series $f(x) = \sum_{n=0}^{\infty} \frac{b_n}{n!} x^n \in \mathbf{Q}\llbracket x \rrbracket$ satisfying: (a) $f$ is a solution of a differential equation; (b) there exists a real number $C > 0$ such that $|b_n| \leq C^n$ and $\operatorname{denom}(b_0, \ldots, b_n) \leq C^n$ for all $n \geq 1$. Show that if $\sum_{n=0}^{\infty} b_n x^n$ is the power series expansion of a rational fraction with rational poles, then $\sum_{n=0}^{\infty} \frac{b_n}{n!} x^n$ is a function $E$.
A function $E$ (with rational coefficients) is a power series $f(x) = \sum_{n=0}^{\infty} \frac{b_n}{n!} x^n \in \mathbf{Q}\llbracket x \rrbracket$ satisfying:
(a) $f$ is a solution of a differential equation;
(b) there exists a real number $C > 0$ such that $|b_n| \leq C^n$ and $\operatorname{denom}(b_0, \ldots, b_n) \leq C^n$ for all $n \geq 1$.
Show that if $\sum_{n=0}^{\infty} b_n x^n$ is the power series expansion of a rational fraction with rational poles, then $\sum_{n=0}^{\infty} \frac{b_n}{n!} x^n$ is a function $E$.