Quotient mean series

The well-known Mathieu series [S.sub.M](r) = [[infinity].summation over (n = 1)] [(2n/[n.sup.2] + [r.sup.2]).sup.2] (r > 0), can be transformed into the form [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where G(n, r) and Q(n, r) denote the Geometric and Quadratic mean of n [member of] N an...

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Bibliographic Details
Published inBanach journal of mathematical analysis Vol. 4; no. 1; p. 87
Main Author Ban, Biserka Drascic
Format Journal Article
LanguageEnglish
Published Springer 01.01.2010
Subjects
Estimation theory
Series
Croatia
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ISSN1735-8787

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Summary:The well-known Mathieu series [S.sub.M](r) = [[infinity].summation over (n = 1)] [(2n/[n.sup.2] + [r.sup.2]).sup.2] (r > 0), can be transformed into the form [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where G(n, r) and Q(n, r) denote the Geometric and Quadratic mean of n [member of] N and r > 0. This connection leads us to the idea to introduce and research the so-called Quotient mean series as a be a generalizations of Mathieu's and Mathieu-type series. We give an integral representation of such series and their alternating variant, together with associated inequalities. Also, special cases of quotient mean series, involving Bessel function of the first kind, have been studied in detail. 2000 Mathematics Subject Classification. Primary 26D15; Secondary 40B05, 40G99. Key words and phrases. Dirichlet-series, quotient mean series, Mathieu series, mean, gamma function, Euler-Maclaurin summation formula, Bessel function of the first kind, Landau estimates, Olenko estimates.
ISSN:1735-8787