Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 1195.33053
Koornwinder, Tom H.
The Askey scheme as a four-manifold with corners. (English)
[J] Ramanujan J. 20, No. 3, 409-439 (2009). ISSN 1382-4090; ISSN 1572-9303/e

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 $$\Bbb R^n_{(q)}:= \{(x_1,\dots,x_n)\in\Bbb R^n\parallel x_{q+1},\dots,x_n\ge 0\} \quad (q=0,1,\dots,n)$$ and gives rise to charts $(U,\varphi)$ such that $\varphi:U\rightarrow \varphi(U)$ is a homeomorphism from an open subset $U$ of $X$ onto an open subset $\varphi(U)$ of some $\Bbb R^n_{(q)}$. 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.
[Marcel G. de Bruin (Haarlem)]

MSC 2000:: *33C45 Orthogonal polynomials and functions of hypergeometric type
33D45 Basic hypergeometric functions and integrals in several variables

Keywords: hypergeometric orthogonal polynomials; Askey scheme; limit relations; three-term recurrence relations; manifold with corners

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]