Quotient mean series
type: Item
instance of: scholarly article
title: Quotient mean series
zbMATH Open document ID: 1195.26038
author: B. D. Ban
published in: Banach Journal of Mathematical Analysis
publication date: 2010-04-27T00:00:00Z
full work available at URL: https://eudml.org/doc/224462
full work available at URL: tex v4 n2 a7.pdf
review text: The quotient mean series \(S^{s,t}_{p,q} \) are defined as \[ S^{s,t}_{p,q}(r_1,r_2)=\sum_{n=1}^{\infty} \frac{(M_2^{[s]}(n,r_1))^t}{(M_2^{[q]}(n,r_2))^p}, \] where \(M_n^{[r]}(a_1, \dots , a_n)\) is a mean of order \(r\) of an \(n\)-tuple \( (a_1, \dots , a_n)\). They are generalizations of the Mathieu series \(S_M(r)= \sum_{i=1}^{\infty} \frac{2n}{(n^2+r^2)^2}\). The author gives an integral representation of the quotient mean series in the following form \[ S^{s,t}_{p,q}(r_1,r_2)= 2^{p/q-t/s}\,\frac{p}{q} \int_0^\infty \int_0^{[u^{1/q}]}\frac{{\mathbf d }_w((w^s+r_1^s)^{t/s})}{(u+r_2^q)^{p/q+1}}\,dw\, du \] where \({\mathbf d}_xa(x)=a(x)+\{x\}a'(x)\). Similar representations for an alternating variant of the quotient mean series and bilateral inequalities are given. Also, special cases of quotient mean series involving Bessel functions of the first kind are considered.
reviewed by: Sanja Varošanec
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 26E60
Mathematics Subject Classification ID: 33C10
Mathematics Subject Classification ID: 40B05
zbMATH DE Number: 5702415
zbMATH Keywords: Dirichlet series
zbMATH Keywords: quotient mean series
zbMATH Keywords: Mathieu series
zbMATH Keywords: mean
zbMATH Keywords: Bessel function of the first kind
MaRDI profile type: Publication
OpenAlex ID: W2013713032
DOI: 10.15352/BJMA/1297117243
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description: scientific article; zbMATH DE number 5702415