  We establish some properties of the zeroes of sums and differences of
contiguous Bessel functions of the first kind. As a byproduct, we also prove
that the zeroes of the derivatives of Bessel functions of the first kind of
different orders are interlaced the same way as the zeroes of Bessel functions
themselves. As a physical motivation, we consider gated graphene nanodots
subject to Berry-Mondragon boundary conditions. We determine the allowed energy
levels and calculate the mean charge at zero temperature. We discuss in detail
its dependence on the gate (chemical) potential.
