The asymptotic zero-counting measure of iterated derivaties of a class of meromorphic functions

Abstract: We give an explicit formula for the logarithmic potential of the asymptotic zero-counting measure of the sequence $\left{\frac{\mathrm{d}<sup>n}{\mathrm{d}z<sup>n}\left(R(z)\exp{T(z)}\right)\right}$. Here, $R(z)$ is a rational function with at least two poles, all of which are distinct, and $T(z)$ is a polynomial. This is an extension of a recent measure-theoretic refinement of P\'olya's Shire theorem for rational functions.