Uniqueness on Difference Operators of Meromorphic Functions of Infinite Order

Abstract

We investigate the uniqueness problems of meromorphic functions and their difference operators by using a new method. It is proved that if a non-constant meromorphic function f shares a non-zero constant and ∞ counting multiplicities with its difference operators Δ_cf(z) and \(\Delta_{c}^{2}f(z)\), then \(\Delta_{c}f(z)\equiv\Delta_{c}^{2}f(z)\). In particular, we give a difference analogue of a result of Jank–Mues–Volkmann. Our method has two distinct features: (i) It converts the relations between functions into the corresponding vectors. This makes it possible to deal with the uniqueness problem by linear algebra and combinatorics. (ii) It circumvents the obstacle of the difference logarithmic derivative lemma for meromorphic functions of infinite order, since this method does not depend on the growth of the functions. Furthermore, the idea in this paper can also be applied to the case for several variables.