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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
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.

References [Enhancements On Off] (What's this?)

  • 1. N. I. Akhiezer, ``The classical moment problem,'' Oliver and Boyd, Edinburgh and London, 1965.
  • 2. R. Askey, Continuous $ q$-Hermite polynomials when $ q>1$, in: ``$ q$-Series and Partitions,'' D. Stanton, (ed.), IMA Vol. Math. Appl., Springer-Verlag, 1989, 151-158. MR 1019849 (90h:33019)
  • 3. G. E. Andrews, R. Askey, and R. Roy, `` Special Functions," Cambridge University Press, Cambridge, 1999. MR 1688958 (2000g:33001)
  • 4. R. Askey and M. E. H. Ismail, Recurrence relations, continued fractions and orthogonal polynomials, Mem. Amer. Math. Soc. 300 (1984). MR 0743545 (85g:33008)
  • 5. R. Askey and R. Roy, More $ q$-beta integrals, Rocky Mountain J. Math. 16 (1986), 365-372. MR 0843057 (87f:33002)
  • 6. C. Berg and J. P. R. Christensen, Density questions in the classical theory of moments, Ann. Inst. Fourier 31 (1981), 99-114.MR 0638619 (84i:44006)
  • 7. C. Berg and M. E. H. Ismail, $ q$-Hermite polynomials and classical orthogonal polynomials, Can. J. Math Vol. 48 (1996), 43-63.MR 1382475 (97b:33020)
  • 8. B. M. Brown and M. E. H. Ismail, A right inverse of the Askey-Wilson operator, Proc. Amer. Math. Soc. Vol. 123 (1995), 2071-2079.MR 1273478 (95i:33019)
  • 9. T. S. Chihara and M. E. H. Ismail, Extremal measures for a system of orthogonal polynomials, Constr. Approx. 9 (1993), 111-119.MR 1198526 (94b:33020)
  • 10. G. Gasper and M. Rahman, Basic hypergeometric series, Cambridge University Press, Cambridge, 1990. MR 1052153 (91d:33034)
  • 11. M. E. H. Ismail, Ladder operators for $ q^{-1}$-Hermite polynomials, C. R. Math. Rep. Acad. Sci. Canada 15 (1993), 261-266. MR 1260071 (95b:33050)
  • 12. M. E. H. Ismail, S. Lin and S. Roan, Bethe Ansatz and $ q$-Sturm-Liouville problems, J. Math. Phys., to appear.
  • 13. M. E. H. Ismail and D. R. Masson, $ q$-Hermite polynomials, biorthogonal rational functions and $ q$-beta integrals, Trans. Amer. Math. Soc 346 (1994), 63-116. MR 1264148 (96a:33022)
  • 14. M. E. H. Ismail and M. Rahman, An inverse to the Askey-Wilson operator, Rocky Mountain J. Math. 32 (2002), 657-678.MR 1934910 (2003m:33022)
  • 15. M. E. H. Ismail and D. Stanton, Classical orthogonal polynomials as moments, Can. J. Math. 49 (1997), 520-542. MR 1451259 (98f:33033)
  • 16. M. E. H. Ismail and D. Stanton, $ q$-integral and moment representations for $ q$-orthogonal polynomials, Can. J. Math. 54 (2002), 709-735. MR 1913916 (2003k:33024)
  • 17. R. Koekoek and R. F. Swarttouw, ``The Askey-scheme of hypergeometric orthogonal polynomials and its $ q$-analogue,'' Report no. 98-17, TU-Delft, 1998.
  • 18. V. E. Korepin, N. M. Bogoliubov, and A. G. Izergin, ``Quantum Inverse Scattering Method and Correlation Functions," Cambridge Univesity Press, Cambridge, 1993. MR 1245942 (95b:81224)
  • 19. J. Shohat and J. D. Tamarkin, The Problem of Moments, revised edition, American Mathematical Society, Providence, 1950. MR 0008438 (5:5c)
  • 20. B. Simon, The classical moment as a selfadjoint finite difference operator, Adv. Math. 137 (1998), 82-203. MR 1627806 (2001e:47020)
  • 21. M. H. Stone, Linear Transformations in Hilbert Space and Their Application to Analysis, American Mathematical Society, Providence, 1932.MR 1451877 (99k:47001)

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Additional Information

Jacob S. Christiansen
Affiliation: Department of Mathematics, University of Copenhagen, Universitetsparken 5, 2100 København, Denmark

Mourad E. H. Ismail
Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816

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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