Algebraic differential independence regarding the Riemann $\mathbf{\zeta }$-function and the Euler $\mathbf{\Gamma }$-function

Abstract

In this paper, we prove that $\mathbf{\zeta }$ cannot be a solution to any nontrivial algebraic differential equation whose coefficients are polynomials in $\mathbf{\Gamma },{\mathbf{\Gamma }}^{(n)}$ and ${\mathbf{\Gamma }}^{(\ell n)}$ over the ring of polynomials in $\mathbf{C}$, where $\ell ,n\ge 1$ are positive integers.