The asymptotic zero-counting measure of iterated derivaties of a class of meromorphic functions (Q1741627)

\textit{G. Pólya} [Math. Zeitschr. 12, 36--60 (1922; JFM 48.0370.02)] proved that the zeros of the iterated derivatives of a meromorphic function \(f\) asymptotically accumulate along the boundary of the Voronoi diagram associated with the set of poles of \(f\). If \(P\) is a polynomial of degree \(d\ge 1\), then its \textit{zero-counting measure} \( \mu\) is a probability measure that assigns mass \(1/d\) to each zero of \(P\), accounting for multiplicity. Theorem 1.1. Let \(f=(P/Q)e^T\), where \(P,Q,T\) are polynomials with \(\mathrm{gcd} (P,Q)=1\), \(\deg P=p\), \(\deg Q=q\ge 2\), \(\deg T=t\ge 1\), \(T(z)=\sum_{k=0}^t d_k z^k\). Furthermore, assume that \(Q\) is monic and that all its zeros \(z_1, \dots, z_q\) are distinct. Then: (i) The sequence of zero-counting measures \((\mu_n)\) of \(f^{(n)}\) converges to a measure \(\mu _S\) with mass \((q-1)/(q-1+t)\). (ii) The sequence of the logarithmic potentials \(L_{\mu_n}(z)\) of \(\mu_n\) diverges pointwise as \(n\to \infty\). (iii) The sequence of the shifted logarithmic potentials \(L_{\mu_n}(z)-(\log n!)/(n(q+t-1)+p)\) converges in \(L_{\mathrm{loc}}^1\) to the subharmonic function \[\Psi(z)=\frac{1}{q+t-1} \Bigl(\max _{1=1,\dots,q} (-\log|z-z_i|)+\log |Q(z)|-\log (|d_t|t)\Bigr).\] (iv) The measure \(\mu_S\) is given by \((2\pi)^{-1} \Delta \Psi(z)\). This theorem generalizes a previous result [\textit{R. Bøgvad} and the author, J. Math. Anal. Appl. 452, No. 1, 312--334 (2017; Zbl 1380.30007)] for the special case that \(f=P/Q\).