We obtain necessary conditions for the non-linear complex differential-difference equations $$w(z + 1)w(z - 1) + a(z)\frac{{{w'}(z)}}{{w(z)}} = R(z,w(z))$$w(z+1)w(z−1)+a(z)w′(z)w(z)=R(z,w(z)) to admit transcendental meromorphic solutions w(z) such that ρ2(w) < 1, where R(z,w(z)) is rational in w(z) with rational coefficients, a(z) is a rational function and ρ2(w) is the hyper-order of w(z). Our results can be seen as the product versions on an equation of another type investigated by Halburd and Korhonen [3]. We also provide an idea which implies that the case of degw(R(z,w)) = 4 in the original proof of [3, Theorem 1.1] can be organized in a short way.

中文翻译：

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非线性复微分差分方程承认亚纯解

我们得到非线性复微分差分方程的必要条件 $$w(z + 1)w(z - 1) + a(z)\frac{{{w'}(z)}}{{w( z)}} = R(z,w(z))$$w(z+1)w(z−1)+a(z)w′(z)w(z)=R(z,w(z) )) 承认超越亚纯解 w(z) 使得 ρ2(w) < 1，其中 R(z,w(z)) 在具有有理系数的 w(z) 中是有理数，a(z) 是有理函数，并且ρ2(w) 是 w(z) 的超阶。我们的结果可以看作是 Halburd 和 Korhonen [3] 研究的另一种类型方程的乘积版本。我们还提供了一个想法，这意味着可以以简短的方式组织 [3，定理 1.1] 的原始证明中 degw(R(z,w)) = 4 的情况。