The MacMahon q-Catalan is Convex

Let n≥ 2 be an integer. We prove the convexity of the so-called MacMahon q-Catalan polynomials C_n(q)=1/[n+1]_q[ 2n n ] _q viewed as functions of q over the entire set of reals. Along the way, several auxiliary properties of the q-Catalan polynomials and intermediate results in the form of inequalities are presented, with the aim to make the paper self-contained. We also include a commentary on the convexity of the generating function for the integer partitions.