Recently, I accidentally stumbled upon the old article [¹] which claimed that the Bessel functions $J_{\nu}$ and $J_{\mu}$ have no common positive zero for any positive real orders $\nu, \mu \in \mathbb{R}^+$.

Recall that the same assertion for $\nu, \mu \in \mathbb{N}$ holds true and usually referred as Bourget’s hypothesis, see p.485 in [²].

However, it is quite easy to see that Bourget’s hypothesis fails, in general, if $\nu, \mu \in \mathbb{R}^+$. Thus, the result of [¹] is already incorrect. The authors of [¹] provided, as a main tool, the following integral criterion for Bourget’s hypothesis:

$J_{\nu}(\lambda) = J_{\mu}(\lambda) = 0$ for some $\lambda, \nu, \mu \in \mathbb{R}^+$ if and only if $\int_0^\lambda \frac{1}{t} J_{\nu}(t) J_{\mu}(t) \, dt = 0$.

This result would be an interesting observation if it were not false. In fact, it is ok in one direction. However, if the integral vanishes, it doesn’t imply the vanishing of $J_\nu$ and $J_\mu$. This can be easily shown numerically by considering, say, $\nu=1$, $\mu=3.2$, and looking for the value of the integral and a common zero (which doesn’t exist) in the interval $(14,15)$.

P.S. Already in 1964 the authors speak about Bourget’s hypothesis as an “old fashion problem”. I feel a bit of shame that, after almost 60 years from [¹], I’m still interested in this very outdated problem, especially in connection with multiplicity of Dirichlet eigenvalues in annuli, see the previous post :-)

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