  We consider properties of binomial series $\sum_{n=0}^\infty a_n
z^{\underline{n}}$, where $z^{\underline{n}}=z(z-1)\cdots(z-n+1)$ and the
convergence of binomial series in the complex domain. The order of growth of
entire and meromorphic solutions of some difference equations represented by
binomial series are discussed. Examples are given. As an application, we
construct a difference Riccati equation possessing a transcendental meromorphic
solution of order $1/2$.
