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

Abstract

For $0\neq x>-1$ let $$\Delta(x)={{\ln \Gamma(x+1)} \over x}.$$ Recently Adell and Alzer proved the complete monotonicity of $\Delta'$ on $(-1,\infty)$ by giving an integral representation of $(-1)^n \Delta^{(n+1)}(x)$ in terms of the Hurwitz zeta function $\zeta(s,a)$. 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 $\Delta^{(n+1)}(x)$ are presented, including its leading asymptotic form as $x \to \infty$.