Linear $q$-difference, difference and differential operators preserving some $\mathcal{A}$-entire functions

TL;DR

This paper investigates how certain linear operators preserve zero distributions of special entire functions, unifying the study of $q$-difference, difference, and differential operators with applications to zero distribution and Hermite-Poulain type theorems.

Contribution

It introduces a unified approach using Rossi's half-plane Borel's theorem to analyze zero distributions under various linear operators with entire coefficients.

Findings

Unified framework for zero distribution analysis of linear operators

Extension of Hermite-Poulain theorem to $q$-difference and difference operators

Existence of infinitely many non-real zeros in certain differential polynomials

Abstract

We apply Rossi's half-plane version of Borel's Theorem to study the zero distribution of linear combinations of $\mathcal{A}$-entire functions (Theorem 1.2). This provides a unified way to study linear $q$-difference, difference and differential operators (with entire coefficients) preserving subsets of $\mathcal{A}$-entire functions, and hence obtain several analogous results for the Hermite-Poulain Theorem to linear finite ($q$-)difference operators with polynomial coefficients. The method also produces a result on the existence of infinitely many non-real zeros of some differential polynomials of functions in certain sub-classes of $\mathcal{A}$-entire functions.