, Volume 2, Issue 1, pp 101-112

Association schemes and quadratic transformations for orthogonal polynomials

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Special cases of the Askey-Wilson polynomials are the eigenmatrices of the classical association schemes. Three constructions on the schemes — multiple polynomial structures, bipartite halves, and antipodal quotients — give quadratic transformations for the polynomials. It is shown that these transformations essentially follow from a quadratic transformation for the Askey-Wilson polynomials. Explicit formulas for the eigenmatrices of three related association schemes are given.

Partially supported by NSF grant DMS: 8500958 and a fellowship from the Sloan Foundation.