Linear differential equations with solutions in weighted Fock spaces

Abstract

This research is concerned with the nonhomogeneous linear complex differential equation $$ f^{(k)}+A_{k-1}f^{(k-1)}+\cdots+A_{1}f'+A_{0}f=A_{k} $$ in the complex plane. In the higher order case, the mutual relations between coefficients and solutions in weighted Fock spaces are discussed, respectively. In particular, sufficient conditions for the solutions of the second order case $$ f"+Af=0 $$ to be in some weighted Fock space are given by Bergman reproducing kernel and coefficient $A$.