Bourget's hypothesis

Remarks

Bourget stated it as a conjecture in the nineteenth century, and Siegel proved it in 1929. The proof is number-theoretic: a common nonzero zero would be a root of a polynomial with rational coefficients, hence algebraic. But Siegel showed that $J_n(\alpha)$ is transcendental, so nonzero, for every nonzero algebraic $\alpha$. Example: $J_0$ and $J_1$ have no common positive zero, which also follows from $J_0' = -J_1$ and the fact that the zeros of $J_0$ are simple. The cases $m = 1$ and $m = 2$ follow from the recurrence relations and the simplicity of nonzero zeros; the general case is the hard part.

Going deeper

For a student

For $m = 1$, a common zero $z_0 \ne 0$ of $J_n$ and $J_{n+1}$ would give $J_n'(z_0) = \frac{n}{z_0} J_n(z_0) - J_{n+1}(z_0) = 0$, a double zero, which is impossible. For $m = 2$ the relation $J_n + J_{n+2} = \frac{2(n+1)}{z} J_{n+1}$ reduces it to the case $m = 1$. For larger $m$ the recurrence writes $J_{n+m}$ as a combination of $J_n$ and $J_{n+1}$ whose coefficients are polynomials in $1/z$ with rational coefficients (the Lommel polynomials). So a common zero must be a root of one of these polynomials, and it is algebraic. Siegel's transcendence theorem for $J_n$ at algebraic points, a starting point of the theory of $E$-functions, then settles it.