J. Aust. Math. Soc.  82 (2007), 369-393

Generalized hypergroups and orthogonal polynomials

Rupert Lasser   GSF—National Research Center   for Environment and Health   Institute of Biomathematics and Biometry   Ingolstädter Landstrasse 1   D–85764 Neuherberg   Germany   lasser@gsf.de Josef Obermaier   GSF—National Research Center   for Environment and Health   Institute of Biomathematics and Biometry   Ingolstädter Landstrasse 1   D–85764 Neuherberg   Germany   josef.obermaier@gsf.de and Holger Rauhut   University of Vienna   Faculty of Mathematics   NuHAG   Nordbergstr. 15   A-1090 Vienna   Austria   holger.rauhut@univie.ac.at

Abstract

The concept of semi-bounded generalized hypergroups (SBG hypergroups) is developed. These hypergroups are more special than generalized hypergroups introduced by Obata and Wildberger and more general than discrete hypergroups or even discrete signed hypergroups. The convolution of measures and functions is studied. In the case of commutativity we define the dual objects and prove some basic theorems of Fourier analysis. Furthermore, we investigate the relationship between orthogonal polynomials and generalized hypergroups. We discuss the Jacobi polynomials as an example.

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