Recently, in the study of unicity problems in complex analysis, I met a problem that can be stated in the following way,

Let $\{m_i\}_{i=0}^{3}$ and $\{n_i\}_{i=0}^{3}$ be eight integers in $\mathbb{Z}\backslash\{0\}$ such that $m_i\neq n_i$ for $0\leq i\leq 3$.
Then $$ R(z):=\sum_{i=0}^{3}(-1)^{i}\binom{3}{i}\frac{1-(\lambda_i z)^{m_i}}{1-(\mu_i z)^{n_i}}-\frac{1-(\lambda_{0}z)^{-m_0}}{1-(\mu_{0}z)^{-n_0}} $$ is not identically to some constant for any non-zero $\lambda_i$ and $\mu_i$.

Of course, this may be checked using the computer in a very direct way. I wondered is there a nice way to discuss it by hands. Thanks in advance.

Recently, in the study of unicity problems in complex analysis, I met a problem that can be stated in the following way,

Let $\{m_i\}_{i=0}^{3}$ and $\{n_i\}_{i=0}^{3}$ be eight integers in $\mathbb{Z}\setminus\{0\}$ such that $m_i\neq n_i$ for $0\leq i\leq 3$.
Then $$ R(z):=\sum_{i=0}^{3}(-1)^{i}\binom{3}{i}\frac{1-(\lambda_i z)^{m_i}}{1-(\mu_i z)^{n_i}}-\frac{1-(\lambda_{0}z)^{-m_0}}{1-(\mu_{0}z)^{-n_0}} $$ is not identically to some constant for any non-zero $\lambda_i$ and $\mu_i$.

Of course, this may be checked using the computer in a very direct way. I wondered is there a nice way to discuss it by hands. Thanks in advance.