On polynomials connected to powers of Bessel functions

The series expansion of a power of the modified Bessel function of the first kind is studied. This expansion involves a family of polynomials introduced by C. Bender et al. New results on these polynomials established here include recurrences in terms of Bell polynomials evaluated at values of the Bessel zeta function. A probabilistic version of an identity of Euler yields additional recurrences. Connections to the umbral formalism on Bessel functions introduced by Cholewinski are established.