Fractional part integral representation for derivatives of a function related to ln Gamma(x+1)

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 $x \to \infty$.

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