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A moment problem and a family of integral evaluations

Christiansen, Jacob S. LU and Ismail, Mourad E.H. (2006) In Transactions of the American Mathematical Society 358(9). p.4071-4097
Abstract

We study the Al-Salam-Chihara polynomials when q > 1. Several solutions of the associated moment problem are found, and the orthogonality relations lead to explicit evaluations of several integrals. The polynomials are shown to have raising and lowering operators and a second order operator equation of Sturm-Liouville type whose eigenvalues are found explicitly. We also derive new measures with respect to which the Ismail-Masson system of rational functions is biorthogonal. An integral representation of the right inverse of a divided difference operator is also obtained.

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author
and
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Al-Salam-Chihara polynomials, Bethe Ansatz equations, Biorthogonal rational functions, Divided difference operators, Indeterminate moment problems, Integral operators, q -Hermite polynomials, Raising and lowering operators
in
Transactions of the American Mathematical Society
volume
358
issue
9
pages
27 pages
publisher
American Mathematical Society (AMS)
external identifiers
  • scopus:33748286736
ISSN
0002-9947
DOI
10.1090/S0002-9947-05-03785-2
language
English
LU publication?
no
id
4b550a00-2bcf-4b66-805b-548dd71ead70
date added to LUP
2025-07-11 11:48:03
date last changed
2025-10-14 12:10:24
@article{4b550a00-2bcf-4b66-805b-548dd71ead70,
  abstract     = {{<p>We study the Al-Salam-Chihara polynomials when q &gt; 1. Several solutions of the associated moment problem are found, and the orthogonality relations lead to explicit evaluations of several integrals. The polynomials are shown to have raising and lowering operators and a second order operator equation of Sturm-Liouville type whose eigenvalues are found explicitly. We also derive new measures with respect to which the Ismail-Masson system of rational functions is biorthogonal. An integral representation of the right inverse of a divided difference operator is also obtained.</p>}},
  author       = {{Christiansen, Jacob S. and Ismail, Mourad E.H.}},
  issn         = {{0002-9947}},
  keywords     = {{Al-Salam-Chihara polynomials; Bethe Ansatz equations; Biorthogonal rational functions; Divided difference operators; Indeterminate moment problems; Integral operators; q -Hermite polynomials; Raising and lowering operators}},
  language     = {{eng}},
  number       = {{9}},
  pages        = {{4071--4097}},
  publisher    = {{American Mathematical Society (AMS)}},
  series       = {{Transactions of the American Mathematical Society}},
  title        = {{A moment problem and a family of integral evaluations}},
  url          = {{http://dx.doi.org/10.1090/S0002-9947-05-03785-2}},
  doi          = {{10.1090/S0002-9947-05-03785-2}},
  volume       = {{358}},
  year         = {{2006}},
}