For $0\neq x>-1$ let $$Δ(x)={{\ln Γ(x+1)} \over x}.$$ Recently Adell and Alzer proved the complete monotonicity of $Δ'$ on $(-1,\infty)$ by giving an integral representation of $(-1)^n Δ^{(n+1)}(x)$ in terms of the Hurwitz zeta function $ζ(s,a)$. We reprove this integral representation in different ways, and then re-express it in terms of fractional part integrals. Special cases then have explicit evaluations. Other relations for $Δ^{(n+1)}(x)$ are presented, including its leading asymptotic form as $x \to \infty$.

15 pages, no figures. Some new references. To appear in Publ. Math. Debrecen