A Difference Version of the Rubel-Yang–Mues-Steinmetz–Gundersen Theorem

In this paper, we give a complete characterization for meromorphic functions that share three distinct values \(a,\,b,\,\infty \) CM, with their difference operator \(\Delta _c f\) or shift \(f(z+c)\). This provides a difference analogue of the corresponding results of Rubel-Yang, Mues-Steinmetz, and Gundersen. In particular, we prove that if an entire function f and its difference derivative \(\Delta _c f\) share three distinct values \(a,\,b,\,\infty \) CM, then \(f\equiv \Delta _c f\). And our results show that the conjecture posed by Chen and Yi in 2013 holds for entire functions, and does not hold for meromorphic functions. Compared with many previous papers, our method 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, but requires the knowledge of linear algebra and combinatorics.