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

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.