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Binomial series and complex difference equations

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 $1/2$.