Two meromorphic functions and are said to share a value or a function if and have the same zeros. One distinguishes the cases whether the values (or functions) are shared counting multiplicities (CM) or ignoring multiplicities (IM). A famous theorem of Nevanlinna says that two meromorphic functions are equal if they share five values IM, and they differ only by a Möbius transformation if they share four values CM [R. Nevanlinna, Acta Math. 48, 367–391 (1926; JFM 52.0323.03)].
There is a vast literature on meromorphic functions sharing values with differential polynomials.
Here, the authors consider the case that a meromorphic function and the shift , where , share values or functions.
It is shown that if is a meromorphic function of finite order, and if and share three values CM, then . In fact, the values may be replaced by periodic meromorphic functions satisfying . If is a deficient value of , then two values or functions suffice. In particular, this is the case for entire . The number of shared values may further be reduced if has also a finite deficient value.
Finally it is shown that if , and share three values CM, where are linearly independent over the reals, then is an elliptic function.