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\(Q\)-classical orthogonal polynomials: A very classical approach. (English) Zbl 0965.33009
The \(q\)-classical polynomials were introduced by Wolfgang Hahn as orthogonal polynomials which have the property that the sequence of their \(L\)-derivatives is also orthogonal. (The \(L\)-derivative is a mixture of a difference operator and the \(q\)-derivative.) By an affine transformation of the variable it is seen that one can restrict oneself to the case when \(L\) is exactly the \(q\)-derivative. These polynomials are of course contained in the Askey scheme of basic hypergeometric orthogonal polynomials. The authors propose an approach to the study of \(q\)-classical polynomials, their classification, and finding of their orthogonality relation by starting from the Sturm-Liouville type equation satisfied by the polynomials, and avoiding “hypergeometrics” as much as possible. Then, in the final section, they present the equivalences with the Askey scheme, including in particular the two pages table from Koekoek and Swarttouw’s compilation [“The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue,” Delft University of Technology, Faculty of Information Technology and Systems, Department of Technical Mathematics and Informatics, Report no. 98-17, 1998; available at http://aw.twi.tudelft.nl/~koekoek/research.html] of the polynomials in the Askey scheme with an indication which of them are \(q\)-classical.

33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
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