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Multivariable \(q\)-Racah polynomials. (English) Zbl 0951.33010

The authors study the multivariable Askey-Wilson (or Koornwinder-Macdonald) polynomials [see T. H. Koornwinder, Askey-Wilson polynomials for root systems of type BC, In: Richards, Donald St. P. (ed.), Hypergeometric functions on domains of positivity, Jack polynomials, and applications. Contemp. Math. 138, 189-204 (1992; Zbl 0797.33014)] in the case that the parameters satisfy a truncation condition such that the orthogonality measure becomes discrete with support on a finite grid. For this choice of the parameters the polynomials can be seen as multivariable \(q\)-Racah polynomials. The discrete orthogonality measure is given as well as some expressions for the normalization constants. The limit case \(q\rightarrow 1\) leading to the multivariable Racah polynomials is treated as well. A special case of interest is the situation that \(|q|=1\), id est \(q\) on the unit circle. In that case there exists a natural parameter domain for which the discrete orthogonality measure, which is complex in general, becomes real-valued and positive.
Reviewer: R.Koekoek (Delft)

MSC:

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

Citations:

Zbl 0797.33014
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References:

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