  In this paper our aim is to present an elementary proof of an identity of
Calogero concerning the zeros of Bessel functions of the first kind. Moreover,
by using our elementary approach we present a new identity for the zeros of
Bessel functions of the first kind, which in particular reduces to some other
new identities. We also show that our method can be applied for the zeros of
other special functions, like Struve functions of the first kind, and modified
Bessel functions of the second kind.
