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

We give an explicit formula for the logarithmic potential of the asymptotic zero-counting measure of the sequence $\left\{\frac{{\mathrm{d}}^n}{\mathrm{d}{z}^n}(R(z)\exp T(z))\right\}$. 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.