## Probabilistic aspects of Al-Salam–Chihara polynomials

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- by Włodzimierz Bryc, Wojciech Matysiak and Paweł J. Szabłowski PDF
- Proc. Amer. Math. Soc.
**133**(2005), 1127-1134 Request permission

## Abstract:

We solve the connection coefficient problem between the Al-Salam–Chihara polynomials and the $q$-Hermite polynomials, and we use the resulting identity to answer a question from probability theory. We also derive the distribution of some Al-Salam–Chihara polynomials, and compute determinants of related Hankel matrices.## References

- Richard Askey and Mourad Ismail,
*Recurrence relations, continued fractions, and orthogonal polynomials*, Mem. Amer. Math. Soc.**49**(1984), no. 300, iv+108. MR**743545**, DOI 10.1090/memo/0300 - W. A. Al-Salam and T. S. Chihara,
*Convolutions of orthonormal polynomials*, SIAM J. Math. Anal.**7**(1976), no. 1, 16–28. MR**399537**, DOI 10.1137/0507003 - Richard Askey,
*Continuous $q$-Hermite polynomials when $q>1$*, $q$-series and partitions (Minneapolis, MN, 1988) IMA Vol. Math. Appl., vol. 18, Springer, New York, 1989, pp. 151–158. MR**1019849**, DOI 10.1007/978-1-4684-0637-5_{1}2 - Christian Berg and Mourad E. H. Ismail,
*$q$-Hermite polynomials and classical orthogonal polynomials*, Canad. J. Math.**48**(1996), no. 1, 43–63. MR**1382475**, DOI 10.4153/CJM-1996-002-4 - Włodzimierz Bryc,
*Stationary random fields with linear regressions*, Ann. Probab.**29**(2001), no. 1, 504–519. MR**1825162**, DOI 10.1214/aop/1008956342 - Saunders MacLane,
*Steinitz field towers for modular fields*, Trans. Amer. Math. Soc.**46**(1939), 23–45. MR**17**, DOI 10.1090/S0002-9947-1939-0000017-3 - Géza Freud.
*Orthogonal Polynomials*. Pergamon Press, Oxford, 1971. - M. E. H. Ismail and D. R. Masson,
*$q$-Hermite polynomials, biorthogonal rational functions, and $q$-beta integrals*, Trans. Amer. Math. Soc.**346**(1994), no. 1, 63–116. MR**1264148**, DOI 10.1090/S0002-9947-1994-1264148-6 - Mourad E. H. Ismail, Mizan Rahman, and Dennis Stanton,
*Quadratic $q$-exponentials and connection coefficient problems*, Proc. Amer. Math. Soc.**127**(1999), no. 10, 2931–2941. MR**1621949**, DOI 10.1090/S0002-9939-99-05017-0 - Mourad E. H. Ismail and Dennis Stanton,
*On the Askey-Wilson and Rogers polynomials*, Canad. J. Math.**40**(1988), no. 5, 1025–1045. MR**973507**, DOI 10.4153/CJM-1988-041-0 - Mourad E. H. Ismail and Dennis Stanton,
*Classical orthogonal polynomials as moments*, Canad. J. Math.**49**(1997), no. 3, 520–542. MR**1451259**, DOI 10.4153/CJM-1997-024-9 - Mourad E. H. Ismail and Dennis Stanton,
*$q$-integral and moment representations for $q$-orthogonal polynomials*, Canad. J. Math.**54**(2002), no. 4, 709–735. MR**1913916**, DOI 10.4153/CJM-2002-027-2 - Mourad E. H. Ismail, Dennis Stanton, and Gérard Viennot,
*The combinatorics of $q$-Hermite polynomials and the Askey-Wilson integral*, European J. Combin.**8**(1987), no. 4, 379–392. MR**930175**, DOI 10.1016/S0195-6698(87)80046-X - C. Krattenthaler,
*Advanced determinant calculus*, Sém. Lothar. Combin.**42**(1999), Art. B42q, 67. The Andrews Festschrift (Maratea, 1998). MR**1701596**

## Additional Information

**Włodzimierz Bryc**- Affiliation: Department of Mathematics, University of Cincinnati, P.O. Box 210025, Cincinnati, Ohio 45221–0025
- Email: Wlodzimierz.Bryc@UC.edu
**Wojciech Matysiak**- Affiliation: Faculty of Mathematics and Information Science, Warsaw University of Technology, pl. Politechniki 1, 00-661 Warszawa, Poland
- Email: wmatysiak@elka.pw.edu.pl
**Paweł J. Szabłowski**- Affiliation: Faculty of Mathematics and Information Science, Warsaw University of Technology, pl. Politechniki 1, 00-661 Warszawa, Poland
- Email: pszablowski@elka.pw.edu.pl
- Received by editor(s): April 22, 2003
- Received by editor(s) in revised form: November 30, 2003
- Published electronically: September 16, 2004
- Additional Notes: This research was partially supported by NSF grant #INT-0332062.
- Communicated by: Carmen C. Chicone
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**133**(2005), 1127-1134 - MSC (2000): Primary 33D45; Secondary 05A30, 15A15, 42C05
- DOI: https://doi.org/10.1090/S0002-9939-04-07593-8
- MathSciNet review: 2117214