Linear differential equations with solutions lying in weighted Fock spaces

Abstract

Sufficient conditions for coefficients of the non-homogeneous linear complex differential equation $f^{\left(k\right)}+A_{k-1}\left(z\right)f^{(k-1)}+\cdots +A_1\left(z\right)f^{\prime }+A_0\left(z\right)f=A_k\left(z\right)$ are found such that all solutions belong to some weighted Fock spaces, where $A_j\left(z\right)$ are entire functions, $j=0,1,\dots ,k$. Furthermore, sufficient conditions for the coefficient $A\left(z\right)$ such that all solutions of the special second order equation $f^{″}+A\left(z\right)f=0$ belong to some weighted Fock spaces are given by the Bergman reproducing kernel, where $A\left(z\right)$ is an entire function.

COMMUNICATED BY:

• H. Boas

Keywords:

• Linear complex differential equation
• entire function
• weighted Fock space

AMS SUBJECT CLASSIFICATIONS:

• 34M10
• 30D35

Acknowledgments

We thank the referees for many important and useful comments. The first author would like to thank Department of Physics and Mathematics, University of Eastern Finland, for providing a good environment during the preparation of this work.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The third author is supported by the National Natural Science Foundation of China (Grant No. 11861023), and the Foundation of Science and Technology project of Guizhou Province of China (Grant No. [2018]5769-05).