The big Picard theorem for polyanalytic functions

Abstract

Let f , g f,g , and h h be polyanalytic in an annular neighborhood A A of a complex number z 0 {z_0} , finite or infinite, such that g g and h h do not have an essential singularity at z 0 {z_0} and g − h g-h is not identically zero on A A . It is shown that if f − g f-g and f − h f-h never vanish on A A , then z 0 {z_0} is not an essential singularity of f f .