Complete monotonicity properties of a function involving the polygamma function

In this paper, we study completete monotonicity properties of the function $f_{a,k}(x)={\psi }^{(k)}(x+a)-{\psi }^{(k)}(x)-\frac{ak!}{x^{k+1}}$, where $a\in (0,1)$ and $k\in {\mathbb{N}}_0$. Specifically, we consider the cases for $k\in \{2n:n\in {\mathbb{N}}_0\}$ and $k\in \{2n+1:n\in {\mathbb{N}}_0\}$. Subsequently, we deduce some inequalities involving the polygamma functions.