Non-existence of entire solution of a type of system of equations. (English) Zbl 07892702
Summary: In this paper, we will prove that the system of differential-difference equations
\[
\begin{cases}
(f(z)f^\prime(z))^n + p_1^2(z)g^m(z + \eta) = Q_1(z), \\
(g(z)g^\prime(z))^n + p_2^2(z)f^m(z + \eta) = Q_2(z),
\end{cases}
\]
has no transcendental entire solution \((f(z)\), \(g(z))\) with \(\rho(f, g) < \infty\) such that \(\lambda(f) < \rho(f)\) and \(\lambda(g) < \rho(g)\), where \(P_1(z)\), \(Q_1(z)\), \(P_2(z)\) and \(Q_2(z)\) are non-vanishing polynomials.
MSC:
| 30D35 | Value distribution of meromorphic functions of one complex variable, Nevanlinna theory |
| 39A45 | Difference equations in the complex domain |
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