On Kaluza's sign criterion for reciprocal power series
type: Item
instance of: scholarly article
title: On Kaluza's sign criterion for reciprocal power series
zbMATH Open document ID: 1260.30002
author: Árpád Baricz
author: Jetro Vesti
author: M. Vuorinen
publication date: 2012-06-08T00:00:00Z
full work available at URL: https://arxiv.org/abs/1010.5337
Mathematics Subject Classification ID: 30B10
Mathematics Subject Classification ID: 33C05
zbMATH DE Number: 6044529
zbMATH Keywords: log-convexity
zbMATH Keywords: hypergeometric functions
MaRDI profile type: Publication
OpenAlex ID: W2951132815
arXiv ID: 1010.5337
DOI: 10.2478/V10062-011-0009-2
describes a project that uses: Digital Library of Mathematical Functions
published in: Annales Universitatis Mariae Curie-Skłodowska. Sectio A. Mathematica
review text: Let \(f(x) = \sum_{n=0}^\infty a_n x^n\) be a Maclaurin series with positive radius of convergence and with positive \(a_n\). Kaluza proved in 1928 that for \(\log\)-convex \((a_n)\) the reciprocal series NEWLINE\[NEWLINE \frac{1}{f(x)} = \frac{1}{a_0} + \sum_{n=1}^\infty b_n x^n NEWLINE\]NEWLINE has non-positive coefficients \(b_n\) (\(n\in\mathbb N\)). In this paper, the sharpness of the \(\log\)-convexity condition is explored and various examples in terms of the Gaussian hypergeometric series are given.
reviewed by: Jürgen Müller
description: scientific article; zbMATH DE number 6044529