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On Picard value problem of some difference polynomials. (English) Zbl 1393.30025

The authors study the value distribution of zeros of certain nonlinear difference polynomials of entire functions of finite order. The following result is obtained: Let \(f\) be a transcendental entire function of finite order, let \(c_1\), \(c_2\) be two nonzero complex numbers such that \(f(z+c_1)\not\equiv f(z+c_2)\), and let \(q\) be a not identically zero polynomial. Then at least one of the functions \(f(z)f(z+c_1)-q(z)\) and \(f(z)f(z+c_2)-q(z)\) has infinitely many zeros.

MSC:

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30D20 Entire functions of one complex variable (general theory)
39A05 General theory of difference equations
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References:

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