A moment problem and a family of integral evaluations
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- by Jacob S. Christiansen and Mourad E. H. Ismail PDF
- Trans. Amer. Math. Soc. 358 (2006), 4071-4097 Request permission
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.References
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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
- MR Author ID: 91855
- Email: ismail@math.ucf.edu
- 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. - © Copyright 2005 American Mathematical Society
- 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
- MathSciNet review: 2219011
Dedicated: This paper is dedicated to Olav Njåstad on the occasion of his seventieth birthday.