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Transactions of the American Mathematical Society

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A moment problem and a family of integral evaluations


Authors: Jacob S. Christiansen and Mourad E. H. Ismail
Journal: Trans. Amer. Math. Soc. 358 (2006), 4071-4097
MSC (2000): Primary 33D45, 44A60; Secondary 47B34
DOI: https://doi.org/10.1090/S0002-9947-05-03785-2
Published electronically: October 31, 2005
MathSciNet review: 2219011
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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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Additional Information

Jacob S. Christiansen
Affiliation: Department of Mathematics, University of Copenhagen, Universitetsparken 5, 2100 København, Denmark
Email: stordal@math.ku.dk

Mourad E. H. Ismail
Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816
Email: ismail@math.ucf.edu

DOI: https://doi.org/10.1090/S0002-9947-05-03785-2
Keywords: Indeterminate moment problems, $q^{-1}$-Hermite polynomials, Al-Salam--Chihara polynomials, biorthogonal rational functions, divided difference operators, raising and lowering operators, Bethe Ansatz equations, integral operators.
Received by editor(s): March 30, 2003
Received by editor(s) in revised form: July 13, 2004
Published electronically: October 31, 2005
Additional Notes: Part of this work was done while the first author was visiting the University of South Florida in Tampa, and he gratefully acknowledges the generous financial support from the private Danish foundation “Travelling Scholarship for Mathematicians” (Rejselegat for Matematikere) and the hospitality of the University of South Florida.
The second author’s research was partially supported by NSF grant DMS 99-70865.
Dedicated: This paper is dedicated to Olav Njåstad on the occasion of his seventieth birthday.
Article copyright: © Copyright 2005 American Mathematical Society

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