Association schemes and quadratic transformations for orthogonal polynomials
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Abstract
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.
Keywords
Explicit Formula Orthogonal Polynomial Association Scheme Classical Association Related Association
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References
- 1.Askey, R., Wilson, J.: A set of orthogonal polynomials that generalize the Racah coefficients or 6-j symbols. SIAM J. Math. Anal.10, 1008–1016 (1979)Google Scholar
- 2.Askey, R., Wilson, J.: Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Am. Math. Soc.319 (1985)Google Scholar
- 3.Bailey, W.: Generalized Hypergeometric Series. Cambridge: Cambridge University Press 1935Google Scholar
- 4.Bannai, E., Bannai, E.: How manyP-polynomial structures can an association scheme have? Europ. J. Comb.,1, 289–298 (1980)Google Scholar
- 5.Bannai, E., Ito, T.: Algebraic Combinatorics 1. Menlo Park, California: Benjamin/Cummings 1984Google Scholar
- 6.Biggs, N.: Algebraic Graph Theory. Cambridge: Cambridge University Press 1974Google Scholar
- 7.Brower, A., Cohen, A., Neumaier, A.: Distance regular graphs (to appear)Google Scholar
- 8.Delsarte, P.: An algebraic approach to the association schemes of coding theory. Philips Research Report Suppl.10 (1973)Google Scholar
- 9.Dunkl, C.: Orthogonal functions on some permutation groups. Proc. Symp. Pure Math.34, 129–147 (1979)Google Scholar
- 10.Foata, D.: Combinatoire des identites sur les polynomes orthgonaux. In: Proceedings of the International Congress of Mathematicians, Warszaw, pp. 1541–1553. Amsterdam-Oxford-New York: North Holland 1983Google Scholar
- 11.Leonard, D.: Orthogonal polynomials, duality, and association schemes. SIAM J. Math. Anal.13, 656–663 (1982)Google Scholar
- 12.Sears, D.: On the transformation theory of basic hypergeometric functions. Proc. London Math. Soc., III. Ser.53, 158–180 (1951)Google Scholar
- 13.Slater, L.: Generalized Hypergeometric Functions. Cambridge: Cambridge University Press 1966Google Scholar
- 14.Stanton, D.: Product formulas forq-Hahn polynomials. SIAM J. Math. Anal.11, 100–107 (1980)Google Scholar
- 15.Stanton, D.: Someq-Krawtchouk polynomials on Chevalley groups, Amer. J. Math.102, 625–662 (1980)Google Scholar
- 16.Stanton, D.: Orthogonal polynomials and Chevalley groups. In: Special Functions: Group Theoretical Aspects and Applications, edited by R. Askey, T.H. Koornwinder, W. Schempp, pp. 87–128. Boston: Reidel 1984Google Scholar
- 17.Viennot, G.: Une theorie combinatoire des polynomes orthogonaux generaux. In: Lecture Notes, Universite du Quebec a Montreal 1983Google Scholar
- 18.Vilenkin, N.: Special Functions and the Theory of Group Representations. Translations American Mathematical Society 22. Providence, Rhode Island: AMS 1968Google Scholar
- 19.Chihara, T.: An Introduction to Orthogonal Polynomials. New York: Gordon and Breach 1978Google Scholar
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