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

TL;DR

This paper provides new integral representations for derivatives of a function related to the logarithm of the Gamma function, including fractional part integrals and explicit evaluations for special cases, enhancing understanding of its properties.

Contribution

It re-derives integral representations of derivatives of the function using different methods and expresses them in terms of fractional part integrals, with explicit evaluations and asymptotic analysis.

Findings

Reproved integral representation of derivatives in alternative ways

Expressed derivatives in terms of fractional part integrals

Derived explicit evaluations for special cases and asymptotic behavior

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 $x \to \infty$.