  We derive a closed formula for the determinant of the Hankel matrix whose
entries are given by sums of negative powers of the zeros of the regular
Coulomb wave function. This new identity applied together with results of
Grommer and Chebotarev allows us to prove a Hurwitz-type theorem about the
zeros of the regular Coulomb wave function. As a particular case, we obtain a
new proof of the classical Hurwitz's theorem from the theory of Bessel
functions that is based on algebraic arguments. In addition, several Hankel
determinants with entries given by the Rayleigh function and Bernoulli numbers
are also evaluated.
