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 Mat. Sb., 2017, Volume 208, Number 3, Pages 28–53 (Mi msb8732)

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

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 $p\ge 3$. 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
$$[\frac{\pi}{p\sin(\pi/p)}, \frac{\pi}{p\sin(\pi/p)\cos(\pi/p)}],$$
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).

Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm8732

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English version:
Sbornik: Mathematics, 2017, 208:3, 335–359

Bibliographic databases:

UDC: 517.518.88+511.331+519.218.2
MSC: Primary 44A60; Secondary 11M32, 30D15, 60J80

Citation: Ch. Berg, R. Szwarc, “Symmetric moment problems and a conjecture of Valent”, Mat. Sb., 208:3 (2017), 28–53; Sb. Math., 208:3 (2017), 335–359

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb8732
• https://doi.org/10.4213/sm8732
• http://mi.mathnet.ru/eng/msb/v208/i3/p28

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This publication is cited in the following articles:
1. Christian Berg, Ryszard Szwarc, “Inverse of Infinite Hankel Moment Matrices”, SIGMA, 14 (2018), 109, 48 pp.
2. I. Bochkov, “Polynomial birth-death processes and the second conjecture of Valent”, C. R. Math. Acad. Sci. Paris, 357:3 (2019), 247–251
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