Linear 𝑞-difference, difference and differential operators preserving some 𝒜-entire functions

Abstract

We apply Rossi’s half-plane version of Borel’s theorem to study the zero distribution of linear combinations of A \mathcal {A} -entire functions (Theorem 1.2). This provides a unified way to study linear q q -difference, difference and differential operators (with entire coefficients) preserving subsets of A \mathcal {A} -entire functions, and hence obtain several analogous results for the Hermite-Poulain theorem to linear finite ( q 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 A \mathcal {A} -entire functions.