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

@inproceedings{Brown1994TheAP,
  title={The Askey-Wilson polynomials and q-Sturm-Liouville problems},
  author={B. M. Brown and William Desmond Evans and Mourad E. H. Ismail},
  year={1994}
}
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. 
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References

SHOWING 1-10 OF 15 REFERENCES
The Combinatorics of q-Hermite polynomials and the Askey - Wilson Integral
On the Askey-Wilson and Rogers Polynomials
The q-shifted factorial (a)n or (a; q)n is and an empty product is interpreted as 1. Recently, Askey and Wilson [6] introduced the polynomials 1.1 where 1.2 and 1.3 We shall refer to these…
Diagonalizaton of certain integral operators II
Symmetry techniques for $q$-series: Askey-Wilson polynomials
We advocate the exploitation of symmetry (recurrence relation) techniques for the derivation of properties associated with families of basic hypergeometric functions, in analogy with the local Lie…
A simple evaluation of Askey and Wilson’s $q$-beta integral
By using the well-known sum of 2 2 series, a simple evaluation is given for Askey and Wilson's ijr-beta type integral ,1 h(x;\)h(x;-l)h(x;)/q')h(x;-xfq') dx
On the Askey-Wilson polynomials
Main properties of the Askey-Wilson polynomials are compactly given on the basis of a generalization of Hahn's approach.
A simple proof of Ramanujan’s summation of the1ψ1
A simple proof by functional equations is given for Ramanujan’s1ψ1 sum. Ramanujan’s sum is a useful extension of Jacobi's triple product formula, and has recently become important in the treatment of…
Basic Hypergeometric Series
Foreword Preface 1. Basic hypergeometric series 2. Summation, transformation, and expansion formulas 3. Additional summation, transformation, and expansion formulas 4. Basic contour integrals 5.…
q-series : their development and application in analysis, number theory, combinatorics, physics, and computer algebra
Found opportunities Classical special functions and L. J. Rogers W. N. Bailey's extension of Roger's work Constant terms Integrals Partitions and $q$-series Partitions and constant terms The hard…
Associated Askey-Wilson polynomials as Laguerre-Hahn orthogonal polynomials
One looks for [formal] orthogonal polynomials satisfying interesting differential or difference relations and equations (Laguerre-Hahn theory). The divided difference operator used here is…
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