Zeros, growth and Taylor coefficients of entire solutions of linear $q$-difference equations

TL;DR

This paper studies the growth, zeros, and Taylor coefficients of entire solutions to linear $q$-difference equations, revealing their asymptotic behaviors and geometric progression patterns.

Contribution

It provides new asymptotic descriptions of zeros and Taylor coefficients of solutions, refining previous growth rate results for these equations.

Findings

Zeros are asymptotic to finitely many geometric progressions

Taylor coefficients exhibit specific asymptotic behavior

Growth rate of solutions is sharpened

Abstract

We consider transcendental entire solutions of linear $q$-difference equations with polynomial coefficients and determine the asymptotic behavior of their Taylor coefficients. We use this to show that under a suitable hypothesis on the associated Newton-Puiseux diagram their zeros are asymptotic to finitely many geometric progressions. We also sharpen previous results on the growth rate of entire solutions.