The big Picard theorem for polyanalytic functions

Abstract:

Let $f,g$, and $h$ be polyanalytic in an annular neighborhood $A$ of a complex number ${z_0}$, finite or infinite, such that $g$ and $h$ do not have an essential singularity at ${z_0}$ and $g-h$ is not identically zero on $A$. It is shown that if $f-g$ and $f-h$ never vanish on $A$, then ${z_0}$ is not an essential singularity of $f$.