Difference independence of the Euler gamma function

In this paper, we established a sharp version of the difference analogue of the celebrated H\"{o}lder's theorem concerning the differential independence of the Euler gamma function $\Gamma$. More precisely, if $P$ is a polynomial of $n+1$ variables in $\mathbb{C}[X, Y_0,\dots, Y_{n-1}]$ such that \begin{equation*} P(s, \Gamma(s+a_0), \dots, \Gamma(s+a_{n-1}))\equiv 0 \end{equation*} for some $(a_0, \dots, a_{n-1})\in \mathbb{C}^{n}$ and $a_i-a_j\notin \mathbb{Z}$ for any $0\leq i<j\leq n-1$, then we have $$P\equiv 0.$$ Our result complements a classical result of algebraic differential independence of the Euler gamma function proved by H\"{o}lder in 1886, and also a result of algebraic difference independence of the Riemann zeta function proved by Chiang and Feng in 2006.