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

TL;DR

This paper extends Pólya's theorem to hyperexponential functions with finite essential singularities, establishing zero distribution laws and microscopic cluster behaviors.

Contribution

It introduces a fixed-scale zero-counting law for hyperexponential functions with arbitrary singularities, extending classical results beyond rational and polynomial-exponential functions.

Findings

Normalized zero-counting measures converge to Voronoi edge measures with weighted atoms at essential singularities.

Microscopic laws of zero clusters follow the reciprocal Marchenko–Pastur and Laguerre Muttalib–Borodin distributions.

Identifies the first sublinear zero layer and its Stokes geometry within essential Voronoi cells.

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.