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Distribution of zeros and shared values of difference operators. (English) Zbl 1236.39021
The authors investigate the distribution of zeros and shared values of the difference operator on meromorphic functions. In particular, they show that if $f$ is a transcendental meromorphic function of finite order with a small number of poles, $c$ is a non-zero complex constant such that $\Delta^k_c f\ne 0$ for $n\ge 2$, and $a$ is a small function with respect to $f$, then $f^n\Delta^k_c f$ equals $a\,(\ne 0,\infty)$ at infinitely many points. Uniqueness of difference polynomials with the same 1-points or fixed points is also proved.

39A70Difference operators
39A05General theory of difference equations
30D35Distribution of values (one complex variable); Nevanlinna theory
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