Completely regular growth of solutions to linear differential equations with exponential polynomial coefficients

Abstract

Consider the linear differential equation $f^{(n)}+A_{n-1}{f}^{(n-1)}+\cdots +A_{0}f=0$, where the coefficients are exponential polynomials. We prove that every finite-order transcendental entire solution has completely regular growth and obtain asymptotic formulas for $\log |f|$ on almost every ray. This answers a question of Heittokangas, Ishizaki, Tohge and Wen. The proof combines logarithmic derivative estimates with asymptotic integration along rays. We express the angular densities of zeros in terms of the indicator. We also extend the result beyond exponential polynomial coefficients under a sectorial Laurent--Puiseux condition.

arXiv comment24 pages, 3 figures. Revised proofs and exposition; includes results on indicators and zero distribution, a Laurent-Puiseux extension, and a Lean 4 formal verification statement