A positive answer for \(3\mathrm{IM}+1\mathrm{CM}\) problem with a general difference polynomial
Summary: In this paper, the \(3\mathrm{IM}+1\mathrm{CM}\) theorem with a general difference polynomial \(L(z, f)\) will be established by using new methods and technologies. Note that the obtained result is valid when the sum of the coefficient of \(L(z, f)\) is equal to zero or not. Thus, the theorem with the condition that the sum of the coefficient of \(L(z, f)\) is equal to zero is also a good extension for recent results. However, it is new for the case that the sum of the coefficient of \(L(z, f)\) is not equal to zero. In fact, the main difficulty of proof is also from this case, which causes the traditional theorem invalid. On the other hand, it is more interesting that the nonconstant finite-order meromorphic function \(f\) can be exactly expressed for the case \(f\equiv-L(z,f)\). Furthermore, the sharpness of our conditions and the existence of the main result are illustrated by examples. In particular, the main result is also valid for the discrete analytic functions.
scientific article; zbMATH DE number 7443964
scientific article