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Sbornik: Mathematics, 2017, Volume 208, Issue 3, Pages 335–359
DOI: https://doi.org/10.1070/SM8732
(Mi sm8732)
 

This article is cited in 3 scientific papers (total in 3 papers)

Symmetric moment problems and a conjecture of Valent

Ch. Berga, R. Szwarcb

a Department of Mathematical Sciences, University of Copenhagen, Denmark
b Institute of Mathematics, University of Wrocław, Poland
References:
Abstract: In 1998 Valent made conjectures about the order and type of certain indeterminate Stieltjes moment problems associated with birth and death processes which have polynomial birth and death rates of degree $p\geqslant 3$. Romanov recently proved that the order is $1/p$ as conjectured. We prove that the type with respect to the order is related to certain multi-zeta values and that this type belongs to the interval
$$ \biggl[\frac{\pi}{p\sin(\pi/p)},\,\frac{\pi}{p\sin(\pi/p)\cos(\pi/p)}\biggr], $$
which also contains the conjectured value. This proves that the conjecture about type is asymptotically correct as $p\to\infty$.
The main idea is to obtain estimates for order and type of symmetric indeterminate Hamburger moment problems when the orthonormal polynomials $P_n$ and those of the second kind $Q_n$ satisfy $P_{2n}^2(0)\sim c_1n^{-1/\beta}$ and $Q_{2n-1}^2(0)\sim c_2 n^{-1/\alpha}$, where $0<\alpha,\beta<1$ may be different, and $c_1$ and $c_2$ are positive constants. In this case the order of the moment problem is majorized by the harmonic mean of $\alpha$ and $\beta$. Here $\alpha_n\sim \beta_n$ means that $\alpha_n/\beta_n\to 1$. This also leads to a new proof of Romanov's Theorem that the order is $1/p$.
Bibliography: 19 titles.
Keywords: indeterminate moment problem, birth and death process with polynomial rates, multi-zeta values.
Funding agency Grant number
National Science Centre (Narodowe Centrum Nauki) 2013/11/B/ST1/02308
R. Szwarc's research was supported by the National Science Centre (NCN), Poland (grant no. 2013/11/B/ST1/02308).
Received: 13.05.2016 and 19.09.2016
Bibliographic databases:
Document Type: Article
UDC: 517.518.88+511.331+519.218.2
MSC: Primary 44A60; Secondary 11M32, 30D15, 60J80
Language: English
Original paper language: Russian
Citation: Ch. Berg, R. Szwarc, “Symmetric moment problems and a conjecture of Valent”, Sb. Math., 208:3 (2017), 335–359
Citation in format AMSBIB
\Bibitem{BerSzw17}
\by Ch.~Berg, R.~Szwarc
\paper Symmetric moment problems and a~conjecture of Valent
\jour Sb. Math.
\yr 2017
\vol 208
\issue 3
\pages 335--359
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\crossref{https://doi.org/10.1070/SM8732}
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\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2017SbMat.208..335B}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85020084967}
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  • https://doi.org/10.1070/SM8732
  • https://www.mathnet.ru/eng/sm/v208/i3/p28
  • This publication is cited in the following 3 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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