  We introduce Omega functions that generalize Euler Gamma functions and study
the functional difference equation they satisfy. Under a natural exponential
growth condition, the vector space of meromorphic solutions of the functional
equation is finite dimensional. We construct a basis of the space of solutions
composed by Omega functions. Omega functions are defined as exponential
periods. They have a meromorphic extension to the complex plane of order $1$
with simple poles at negative integers. They are characterized by their growth
property on vertical strips and their functional equation. This generalizes
Wielandt's characterization of Euler Gamma function. We also introduce
Incomplete Omega functions that play an important role in the proofs.
