Uniqueness of meromorphic functions sharing values with their shifts

Abstract

Shared value problems related to a meromorphic function f (z) and its shift f (z + c), where c ∈ ℂ, are studied. It is shown, for instance, that if f (z) is of finite order and shares at least three values counting multiplicities with its shift f (z + c), then f is a periodic function with period c. The assumption on the order of f can be dropped if f shares two shifts in different directions. More precisely, it is proved that if c 1 and c 2 are complex constants which are linearly independent over the real numbers, and f (z) is any non-constant meromorphic function sharing three values counting multiplicities with the shifted functions f (z + c 1) and f (z + c 2), then f is an elliptic function with periods c 1 and c 2. This can be seen as a new way of characterizing elliptic functions.