  We consider the zeros distributions on the derivatives of difference
polynomials of meromorphic functions, and present some results which can be
seen as the discrete analogues of Hayman conjecture \cite{hayman1}, also partly
answer the question given in \cite[P448]{luolin}. We also investigate the
uniqueness problems of difference-differential polynomials of entire functions
sharing one common value. These theorems improve the results of Luo and
Lin\cite{luolin} and some results of present authors \cite{liuliucao}.
