Zero asymptotics for successive derivatives of hyperexponential functions with finite essential singularities

Abstract

P\'olya's shire theorem identifies the final set of zeros of successive derivatives of an arbitrary meromorphic function with at least one pole with the Voronoi diagram of its finite poles. We prove a fixed-scale zero-counting law for hyperexponential functions $f=(P/Q)\exp(S/T)$, allowing ordinary poles and finite essential singularities of arbitrary order and position, thus extending P\'olya's picture beyond the rational, polynomial-exponential, and one-dimensional finite-essential-singularity settings. After the forced singular factors are removed from the numerator of $f^{(n)}$, the normalized zero-counting measures converge in the original $z$-plane to the classical Voronoi edge measure generated by all finite singular sites, augmented by explicitly weighted atoms at the finite essential singularities, which thereby enter P\'olya's picture both as Voronoi sites and as sources of linear-size zero clusters. If $S/T$ has a nonconstant polynomial part, the complementary mass escapes to infinity. We determine the microscopic laws of these clusters, obtaining the reciprocal Marchenko--Pastur law for simple poles of $S/T$ and higher-order multiple-Laguerre, equivalently Laguerre Muttalib--Borodin, limits for higher-order poles. Finally, inside essential Voronoi cells we identify the first sublinear zero layer, including its Stokes geometry, densities, and final-set consequences away from transition loci.