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The Askey scheme as a four-manifold with corners. (English) Zbl 1195.33053

The author gives a beautiful treatment of the Askey-Wilson scheme with the limit relations between its members. Use is made of dilation/translation of parameters and subsequent re-parametrization (to lead to polynomials orthogonal in their parameters when these are non-negative) and in such a manner that restriction of one or more of these parameters to zero leads to orthogonal polynomials lower in the Askey scheme (i.e., limit transitions are seen as point evaluation in the parameter space).

In this way, it is possible to give a geometrical description as a manifold with corners. This type of manifold $X$ can be described by

${ℝ}_{\left(q\right)}^{n}:=\left\{\left({x}_{1},\cdots ,{x}_{n}\right)\in {ℝ}^{n}\parallel {x}_{q+1},\cdots ,{x}_{n}\ge 0\right\}\phantom{\rule{1.em}{0ex}}\left(q=0,1,\cdots ,n\right)$

and gives rise to charts $\left(U,\varphi \right)$ such that $\varphi :U\to \varphi \left(U\right)$ is a homeomorphism from an open subset $U$ of $X$ onto an open subset $\varphi \left(U\right)$ of some ${ℝ}_{\left(q\right)}^{n}$.

Each point of this manifold can be associated with a system of orthogonal polynomials.

The Askey scheme is then covered in five local charts; the Racah manifold is covered with three charts and the Wilson one with two.

Finally, the paper concludes with a section discussing the results found and giving an inroad to work to be done in the future.

##### MSC:
 33C45 Orthogonal polynomials and functions of hypergeometric type 33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
##### References:
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