Bourget's Hypothesis

When $n$ is an Integer $\geq 0$, then $J_n(z)$ and $J_{n+m}(z)$ have no common zeros other than at $z=0$ for $m$ an Integer $\geq 1$, where $J_n(z)$ is a Bessel Function of the First Kind. The theorem has been proved true for $m=1$ 2, 3, and 4.

References

Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966.