Title: On a Neumann-type series for modified Bessel functions of the first kind,
Abstract: In this paper, we are interested in a Neumann-type series for modified Bessel
functions of the first kind which arises in the study of Dunkl operators
associated with dihedral groups and as an instance of the Laguerre semigroup
constructed by Ben Said-Kobayashi-Orsted. We first revisit the particular case
corresponding to the group of square-preserving symmetries for which we give
two new and different proofs other than the existing ones. The first proof uses
the expansion of powers in a Neumann series of Bessel functions while the
second one is based on a quadratic transformation for the Gauss hypergeometric
function and opens the way to derive further expressions when the orders of the
underlying dihedral groups are powers of two. More generally, we give another
proof of De Bie \& al formula expressing this series as a $\Phi_2$-Horn
confluent hypergeometric function. In the course of proving, we shed the light
on the occurrence of multiple angles in their formula through elementary
symmetric functions, and get a new representation of Gegenbauer polynomials.