Some Results on a Class of Polynomials Related to Convolutions of the Catalan Sequence

Motivated by the search of the singular values of Jordan blocks, in a previous paper (Capparelli and Maroscia in Med J Math 10:1609-1630, 2013) we studied, among other things, a family of monic polynomials with integer coefficients that turned out to be linked to convolutions of the sequence of Catalan numbers. In the present paper, we continue the study of these polynomials and prove, in particular, the irreducibility of an infinite subset of them. As an interesting byproduct, we also obtain a simple rational function in two variables which can be naturally thought of as the generating function of the Catalan number sequence and all its convolutions.