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 the Bessel function $$J_0(x) \stackrel{\text{def}}{=} \sum_{n=0}^{\infty} \frac{(-1)^n}{(n!)^2} \left(\frac{x}{2}\right)^{2n}$$ is a function $E$ such that $\widehat{J}_0(x)$ satisfies the equation $(1 + x^2)\widehat{J}_0(x)^2 = 1$. Deduce that $J_0(x)$ is not an exponential polynomial.
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 the Bessel function
$$J_0(x) \stackrel{\text{def}}{=} \sum_{n=0}^{\infty} \frac{(-1)^n}{(n!)^2} \left(\frac{x}{2}\right)^{2n}$$
is a function $E$ such that $\widehat{J}_0(x)$ satisfies the equation $(1 + x^2)\widehat{J}_0(x)^2 = 1$. Deduce that $J_0(x)$ is not an exponential polynomial.