Extremal values of multiple gamma and sine functions
type: Item
instance of: scholarly article
title: Extremal values of multiple gamma and sine functions
zbMATH Open document ID: 1190.33022
author: Kazuhiro Onodera
published in: Advances in Mathematics
publication date: 2010-05-26T00:00:00Z
review text: Barnes's multiple gamma function \(\Gamma_{r}(x)\) is defined by \[ \Gamma_{r}(x)=\exp\Big(\frac{\partial}{\partial s}\zeta_{r}(s,\,x)\Big|_{s=0}\Big) \;\;\;(x>0) \] where \(\zeta_{r}(s,\,x)\) is the multiple Hurwitz zeta function defined as \[ \zeta_{r}(s,\,x)=\sum_{n_{1},\dots,n_{r}=0}^{\infty}\frac{1}{(n_{1}+\dots + n_{r}+x)^{s}}\;\;\;\;(\text{Re} (s)>r). \] The multiple sine function \(S_{r}(x)\) is defined by \[ S_{r}(x)=\Gamma_{r}(x)^{-1}\,\Gamma_{r}(r-x)^{(-1)^{r}}\;\;\;\;(0<x<r). \] The author studies the extremal values of the functions \(\Gamma_{r}(x)\) and \(S_{r}(x)\) in the fundamental intervals, shows the number and locations of the extremal points and also proves that all the local maximum and minimum values are greater and less than one, respectively.
reviewed by: Stamatis Koumandos
Mathematics Subject Classification ID: 33E20
Mathematics Subject Classification ID: 33B15
zbMATH DE Number: 5712548
zbMATH Keywords: multiple gamma function
zbMATH Keywords: multiple sine function
zbMATH Keywords: Stirling modular form
zbMATH Keywords: extremal values
MaRDI profile type: Publication
full work available at URL: j.aim.2009.12.019
OpenAlex ID: W1984419983
cites work: Sums involving the Hurwitz zeta function
cites work: Division values of multiple sine functions
cites work: Values of the double sine function
cites work: Gamma factors and Plancherel measures
cites work: The central value of the triple sine function
cites work: Multiple sine functions
cites work: Generalized Hölder's theorem for multiple gamma function
cites work: BEHAVIOR OF SOME GENERALIZED MULTIPLE SINE FUNCTIONS
cites work: Weierstrass product representations of multiple gamma and sine functions
cites work: Central values of generalized multiple sine functions
DOI: 10.1016/J.AIM.2009.12.019
Recommended article: Special values of multiple gamma functions
Recommended article: Multiple gamma functions, multiple sine functions, and Appell's \(O\)-functions
Recommended article: On the multiple gamma-functions
Recommended article: EXTREMAL VALUES OF DOUBLE AND TRIPLE TRIGONOMETRIC FUNCTIONS
Recommended article: Multiple gamma and related functions
Recommended article: Multiple Gamma Functions and Their Applications
Recommended article: On the special functions higher than the multiple gamma-functions
Recommended article: Bounding the gamma function in terms of the trigonometric and exponential functions
Recommended article: Multiple gamma functions and \(L\)-functions
Recommended article: Q4438538
description: scientific article; zbMATH DE number 5712548