Partition functions of two-dimensional Coulomb gases with circular root- and jump-type singularities

In this paper, we study the random polynomial $p_n(ρ):=\prod_{j=1}^n (|z_j|-ρ)$, where the points $\{z_j\}_{j=1}^n$ are the eigenvalue moduli of random normal matrices with a radially symmetric potential. We establish precise large $n$ asymptotic expansions for the moment generating function \[ \mat