Date: Mon, 1 Jul 1996
From: Rene Swarttouw rene@cs.vu.nl and Tom Koornwinder thk@fwi.uva.nl
Topic: A computer implementation of the Askey-Wilson scheme

This is an update of the information on the project "A computer implementation of the Askey-Wilson scheme" (see OP-SF Net 3.1, topic #16 and OP-SF Net 3.3, topic #11). During the period January - June 1996, René Swarttouw has worked on this project at RIACA in Eindhoven, The Netherlands. Tom Koornwinder was an adviser for this project. Starting July 1, 1996 René Swarttouw is back at the Free University, Amsterdam, and his main task will be in another direction. The project will be continued in a lower gear by Swarttouw and Koornwinder until new funds have been raised and new staff has been hired for a follow-up of the project.

To see the latest output of the project, it is now best to visit René Swarttouw's home page and then follow the link to the Askey-Wilson scheme project. You will find there the interactive book form of the Koekoek-Swarttouw report The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, both in a hypertex version and in a Netscape version.

You will also find there a link to CAOP, a package for calculating formulas for orthogonal polynomials belonging to the Askey-scheme by Maple.

With the present version users can compute recurrence relations, differential and difference equations or make a plot of every polynomial in the Askey scheme, without having Maple installed on their own computer. It is also possible to multiply the polynomial by a scaling function, to change the argument and to give values to the parameters by filling out a form, before doing the calculation. As an extra option the user can choose the layout of the output: prettyprint, lineprint or LaTeX. The latter two options make it possible to insert the output in another Maple worksheet respectively in a LaTeX document by a simple mouse-action. Furthermore there are some help pages available for users who are not familiar with Maple. They can be viewed simultaneously while filling in the form.

Finally, Tom Koornwinder and René Swarttouw are writing together a procedure in Maple, called rec2ortho, which uses the coefficients of a three-term recurrence relation for orthogonal polynomials as input, and which returns the explicit system of OP's solving these recurrence relations, provided it belongs to the Askey scheme or q-Askey scheme. Eventually, this should also include associated OP's and other explicit OP's which are not in the (q-)Askey scheme. At the moment, the procedure only covers the OP's in the Askey scheme (q = 1) up to the level of 2 parameters (Jacobi, Meixner, etc.). For optimal usage of the procedure you need Maple V, Release 4 (already available for Windows and Macintosh, coming out for Sun workstation soon). The procedure rec2ortho, as well as a test file rec2orthotest, can be obtained by anonymous ftp from

ftp.fwi.uva.nl
cd pub/mathematics/reports/Analysis/koornwinder/rec2ortho.dir

or directly by clicking their names above.

René Swarttouw rene@cs.vu.nl and Tom Koornwinder thk@fwi.uva.nl welcome questions, comments and suggestions about this project.

See also Extended version of Askey-Wilson-scheme report