Inequalities involving \(q\)-analogue of multiple psi functions
type: Item
instance of: scholarly article
title: Inequalities involving \(q\)-analogue of multiple psi functions
zbMATH Open document ID: 1450.33004
author: Sourav Das
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2020-08-03T00:00:00Z
review text: One can define the \(q\)-analogue of the multiple Gamma functions formally the very same way as the multiple Gamma functions are defined: \[\Gamma_n(z;q)=G_n(z;q)^{-1^{n-1}}.\] Here \[\begin{aligned} \log G_{n}(z+1 ; q) &=-\left(\begin{array}{l} z \\ n \end{array}\right) \log (1-q)-\sum_{i=1}^{\infty}\left(\begin{array}{c} -k \\ n-1 \end{array}\right) \log \left(1-q^{z+k}\right) \\ &+\sum_{j=0}^{n-1} G_{n, j}(z)\left\{\sum_{k=1}^{\infty} k^{j} \log \left(1-q^{k}\right)\right\} \end{aligned},\] where the \(G_{n,j}(z)\) polynomials are defined as \[\left(\begin{array}{l} z-u \\ n-1 \end{array}\right)=\sum_{j=0}^{n-1} G_{n, j}(z) u^{j}.\] Taking logarithmic derivative in the definition of \(\Gamma_n(z;q)\), one arrives at the definition of the \(q\)-analogue of the multiple polygamma functions: \[\Psi_{n}^{(m)}(z ; q)=(\ln q)^{m+1} \sum_{k=1}^{\infty}\left(\begin{array}{c} n+k-2 \\ n-1 \end{array}\right) \frac{k^{m} q^{k z}}{1-q^{k}}, \quad 0<q<1, \quad m \geq n.\] The author derives convexity properties for these functions. For example, it is proven that \(\Psi_{n}^{(m)}(x ; q)\) is convex (concave) on \((0, \infty)\) if \(m\) is odd (even) for \(0<q<1\). Numerous other statements regarding superadditivity of the \(\Psi_{n}^{(m)}(x ; q)\) functions are given, too.
Mathematics Subject Classification ID: 33B15
Mathematics Subject Classification ID: 26D07
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 7226790
zbMATH Keywords: superadditivity
zbMATH Keywords: convexity
zbMATH Keywords: multiple \(q\)-polygamma functions
reviewed by: István Mező
MaRDI profile type: Publication
full work available at URL: crmath.44
OpenAlex ID: W3041848522
cites work: Sharp inequalities for the digamma and polygamma functions
cites work: Sub- and superadditive properties of Euler's gamma function
cites work: A subadditive property of the gamma function.
cites work: Monotonicity properties of \(q\)-digamma and \(q\)-trigamma functions
cites work: Multiple Gamma Functions and Their Applications
cites work: The \(q\)-deformed gamma function and \(q\)-deformed polygamma function
cites work: The additivity of polygamma functions
cites work: Q3622978
cites work: On Barnes' multiple zeta and gamma functions
cites work: Q3021451
cites work: Q4343381
cites work: Q4184934
DOI: 10.5802/CRMATH.44
description: scientific article; zbMATH DE number 7226790