Fractional part integral representation for derivatives of a function related to \(\ln\Gamma(x + 1)\) (Q2915404)

Let NEWLINE\[NEWLINE\Delta(x)=\frac{\ln \Gamma(x+1)}{x}, \quad 0\neq x>-1.NEWLINE\]NEWLINE The author gives a new proof of a result due to \textit{J. A. Adell} and \textit{H. Alzer} [Publ. Math. 78, No. 2, 443--448 (2011; Zbl 1240.26018)] concerning an integral representation of \((-1)^n\,\Delta^{(n+1)}(x)\) in terms of the Hurwitz zeta function \(\zeta(s, a)\). The author re-expresses the same integral representation in terms of fractional part integrals and gives explicit evaluations of special cases. Other relations for \(\Delta^{(n+1)}(x)\) are presented, including its leading asymptotic form as \(x\to \infty\).