Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-30T19:23:30.105Z Has data issue: false hasContentIssue false
  • English
  • Français

The Askey–Wilson polynomials and q-Sturm–Liouville problems*

Published online by Cambridge University Press:  24 October 2008

B. Malcolm Brown ,
W. Desmond Evans  and
Mourad E. H. Ismail
B. Malcolm Brown
Affiliation:
Department of Computing Mathematics, University of Wales College at CardiffCardiff, Wales CF2 4YN
W. Desmond Evans
Affiliation:
School of Mathematics, University of Wales College at Cardiff, Cardiff, Wales CF4 2YN
Mourad E. H. Ismail
Affiliation:
Department of Mathematics, University of South Florida, Tampa, FL 33620, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

We find the adjoint of the Askey–Wilson divided difference operator with respect to the inner product on L2(–1, 1, (1– x2)½dx) defined as a Cauchy principal value and show that the Askey-Wilson polynomials are solutions of a q-Sturm–Liouville problem. From these facts we deduce various properties of the polynomials in a simple and straightforward way. We also provide an operator theoretic description of the Askey-Wilson operator.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 119 , Issue 1 , January 1996 , pp. 1 - 16
Copyright
Copyright © Cambridge Philosophical Society 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Andrews, G. E.. q-Series: their development and application in analysis, number theory, combinatorics, physics, and computer algebra, CBMS Regional Conference Series 66 (American Mathematical Society, 1986).CrossRefGoogle Scholar
[2]Andrews, G. E. and Askey, R. A.. A simple proof of Ramanujan's summation of the 1ψ1. Aequationes Math. 18 (1978), 333–337.CrossRefGoogle Scholar
[3]Andrews, G. E. and Askey, R. A.. Classical orthogonal polynomials. In Polynomes Orthogonaux et applications (eds. Bresinski, C. et al. ). Lecture Notes in Mathematics (Springer-Verlag, 1984), pp. 36–63.Google Scholar
[4]Askey, R. A.. Divided difference operators and classical orthogonal polynomials. Rocky Mountain J. Math. 19 (1969), 33–37.Google Scholar
[5]Askey, R. A. and Wilson, J. A.. Some basic hypergeometric polynomials that generalize Jacobi polynomials. Memoirs Amer. Math. Soc. 319 (1985).Google Scholar
[6]Atakishiyev, N. M. and Suslov, S. K.. On the Askey-Wilson polynomials. Constructive Approximation 8 (1992), 363–369.CrossRefGoogle Scholar
[7]Gasper, G. and Rahman, M.. Basic Hypergeometric Series (Cambridge University Press, 1990).Google Scholar
[8]Ismail, M. E. H.. A simple proof of Ramanujan's 1ψ1 sum. Proc. Amer. Math. Soc. 63 (1978), 185–186.Google Scholar
[9]Ismail, M. E. H.. The Askey-Wilson operator and summation theorems. Contemporary Math. (to appear).Google Scholar
[10]Ismail, M. E. H., Rahman, M. and Zhang, R.. Diagonalization of certain integral operators II. J. Comp. Appl. Math, (to appear).Google Scholar
[11]Ismail, M. E. H., and Stanton, D.. On the Askey-Wilson and Rogers polynomials. Canad. J. Math. 40 (1988), 1025–1045.CrossRefGoogle Scholar
[12]Ismail, M. E. H., Stanton, D. and Viennot, G.. The combinatorics of the q–Hermite polynomials and the Askey-Wilson integral. European J. Combinatorics 8 (1987), 379–392.CrossRefGoogle Scholar
[13]Kalnins, E. G. and Miller, W.. Symmetry techniques for q–series: Askey-Wilson polynomials. Rocky Mountain J. Math. 19 (1989), 223–230.CrossRefGoogle Scholar
[14]Labelle, G.. Tableau d'askey. In Polynomes orthogonaux et applications (eds. Breziniski, C. et al. ). Lecture Notes in Mathematics 1171 (Springer-Verlag, 1984), pp. 36–37.Google Scholar
[15]Magnus, A. P.. Associated Askey-Wilson polynomials as Laguerre-Hahn orthogonal polynomials. In Orthogonal polynomials and their applications (eds. Alfaro, M. et al. ). Lecture Notes in Mathematics 1329 (Springer-Verlag, 1988), pp. 261–278.CrossRefGoogle Scholar
[16]Rahman, M.. A simple evaluation of Askey and Wilson's q–beta integral. Proc. Amer. Math. Soc. 92 (1984), 413–417.Google Scholar
[17]Rainville, E. D.. Special Functions (Chelsea, 1971).Google Scholar
[18]Wilson, J. A.. Three term contiguous relations and some new orthogonal polynomials. In Fade and rational approximation (eds. Saff, E. B. and Varga, R. S.). (Academic Press. 1977). pp. 227–232.CrossRefGoogle Scholar