  The use of the umbral formalism allows a significant simplification of the
derivation of sum rules involving products of special functions and
polynomials. We rederive in this way known sum rules and addition theorems for
Bessel functions. Furthermore, we obtain a set of new closed form sum rules
involving various special polynomials and Bessel functions. The examples we
consider are relevant for applications ranging from plasma physics to quantum
optics.
