Common zeros of two Bessel functions

Abstract:

There is a theorem that two Bessel functions ${J_\mu }(x)$ and ${J_\nu }(x)$ can have no common positive zeros if $\mu$ is an integer and $\nu = \mu + m$ where m is an integer, but this does not preclude the possibility that for unrestricted real positive $\mu$ and $\nu$ not differing by an integer, the two functions ${J_\mu }(x)$ and ${J_\nu }(x)$ can have common zeros. An example is found where two such functions have two positive zeros in common.