In:
Matematicheskii Sbornik, Steklov Mathematical Institute, Vol. 208, No. 3 ( 2017), p. 28-53
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
$$ [\frac{\pi}{p\sin(\pi/p)}, \frac{\pi}{p\sin(\pi/p)\cos(\pi/p)}], $$
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.
Type of Medium:
Online Resource
ISSN:
0368-8666
,
2305-2783
Language:
Russian
Publisher:
Steklov Mathematical Institute
Publication Date:
2017
detail.hit.zdb_id:
203247-8
SSG:
17,1
