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A question by T.S. Chihara about shell polynomials and indeterminate moment problems

Berg, Christian and Christiansen, Jacob Stordal LU (2011) In Journal of Approximation Theory 163(10). p.1449-1464
Abstract

The generalized Stieltjes-Wigert polynomials depending on parameters 0≤;p<1 and 0<q<1 are discussed. By removing the mass at zero of an N-extremal solution concentrated in the zeros of the D-function from the Nevanlinna parametrization, we obtain a discrete measure μM, which is uniquely determined by its moments. We calculate the coefficients of the corresponding orthonormal polynomials (PnM). As noticed by Chihara, these polynomials are the shell polynomials corresponding to the maximal parameter sequence for a certain chain sequence. We also find the minimal parameter sequence, as well as the parameter sequence corresponding to the generalized Stieltjes-Wigert polynomials, and compute the value of related continued fractions.... (More)

The generalized Stieltjes-Wigert polynomials depending on parameters 0≤;p<1 and 0<q<1 are discussed. By removing the mass at zero of an N-extremal solution concentrated in the zeros of the D-function from the Nevanlinna parametrization, we obtain a discrete measure μM, which is uniquely determined by its moments. We calculate the coefficients of the corresponding orthonormal polynomials (PnM). As noticed by Chihara, these polynomials are the shell polynomials corresponding to the maximal parameter sequence for a certain chain sequence. We also find the minimal parameter sequence, as well as the parameter sequence corresponding to the generalized Stieltjes-Wigert polynomials, and compute the value of related continued fractions. The mass points of μM have been studied in recent papers of Hayman, Ismail-Zhang and Huber. In the special case of p=q, the maximal parameter sequence is constant and the determination of μM and (PnM) gives an answer to a question posed by Chihara in 2001.

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publishing date
type
Contribution to journal
publication status
published
subject
keywords
Chain sequences, Orthogonal polynomials, Q-series, Stieltjes-Wigert polynomials
in
Journal of Approximation Theory
volume
163
issue
10
pages
16 pages
publisher
Academic Press
external identifiers
  • scopus:80051568917
ISSN
0021-9045
DOI
10.1016/j.jat.2011.05.002
language
English
LU publication?
no
id
b2ea7192-eaeb-47f3-8a56-5c25b2192266
date added to LUP
2025-07-11 12:13:36
date last changed
2025-10-14 11:05:10
@article{b2ea7192-eaeb-47f3-8a56-5c25b2192266,
  abstract     = {{<p>The generalized Stieltjes-Wigert polynomials depending on parameters 0≤;p&lt;1 and 0&lt;q&lt;1 are discussed. By removing the mass at zero of an N-extremal solution concentrated in the zeros of the D-function from the Nevanlinna parametrization, we obtain a discrete measure μM, which is uniquely determined by its moments. We calculate the coefficients of the corresponding orthonormal polynomials (PnM). As noticed by Chihara, these polynomials are the shell polynomials corresponding to the maximal parameter sequence for a certain chain sequence. We also find the minimal parameter sequence, as well as the parameter sequence corresponding to the generalized Stieltjes-Wigert polynomials, and compute the value of related continued fractions. The mass points of μM have been studied in recent papers of Hayman, Ismail-Zhang and Huber. In the special case of p=q, the maximal parameter sequence is constant and the determination of μM and (PnM) gives an answer to a question posed by Chihara in 2001.</p>}},
  author       = {{Berg, Christian and Christiansen, Jacob Stordal}},
  issn         = {{0021-9045}},
  keywords     = {{Chain sequences; Orthogonal polynomials; Q-series; Stieltjes-Wigert polynomials}},
  language     = {{eng}},
  number       = {{10}},
  pages        = {{1449--1464}},
  publisher    = {{Academic Press}},
  series       = {{Journal of Approximation Theory}},
  title        = {{A question by T.S. Chihara about shell polynomials and indeterminate moment problems}},
  url          = {{http://dx.doi.org/10.1016/j.jat.2011.05.002}},
  doi          = {{10.1016/j.jat.2011.05.002}},
  volume       = {{163}},
  year         = {{2011}},
}