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 ${\lbrace (d^n/dz^n) (R(z) \mathrm{exp}T(z))\rbrace}^{\infty}_{n=1}$. 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 Pólya’s Shire theorem for rational functions.

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Received 4 October 2017

Accepted 5 August 2018

Published 3 May 2019