On algebraic differential equations concerning the Riemann-zeta function and the Euler-gamma function

In this paper, we prove that $\zeta$ is not a solution of any non-trivial algebraic differential equation whose coefficients are polynomials in $\Gamma, \Gamma^{(n)}, \Gamma^{(l)}$ over the ring of polynomials in $\mathbb{C}$, $l>n\geq 1$ are positive integers. We extended the result that $\zeta$ does not satisfy any non-trivial algebraic differential equation whose coefficients are polynomials in $\Gamma, \Gamma', \Gamma''$ over the field of complex numbers, which is proved by Li and Ye[7].