Fractional part integral representation for derivatives of a function related to \(\ln\Gamma(x + 1)\)
type: Item
instance of: scholarly article
zbMATH Open document ID: 1265.33004
DOI: 10.5486/PMD.2012.5039
author: Mark W. Coffey
published in: Publicationes Mathematicae Debrecen
publication date: 2012-09-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/1101.4257
Mathematics Subject Classification ID: 33B15
Mathematics Subject Classification ID: 11M35
Mathematics Subject Classification ID: 11Y60
zbMATH DE Number: 6083237
zbMATH Keywords: gamma function
zbMATH Keywords: digamma function
zbMATH Keywords: polygamma function
zbMATH Keywords: Hurwitz zeta function
zbMATH Keywords: Riemann zeta function
zbMATH Keywords: fractional part
zbMATH Keywords: integral representation
MaRDI profile type: Publication
arXiv ID: 1101.4257
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title: Fractional part integral representation for derivatives of a function related to \(\ln\Gamma(x + 1)\)
review text: 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 \(x\to \infty\).
reviewed by: Stamatis Koumandos
description: scientific article; zbMATH DE number 6083237