Three lectures on Hypergroups Delhi, December 1995

Abstract

LECTURE 1. Cosines, Legendre polynomials, and Bessel functions are examples of eigenfunctions of Sturm-Liouville problems which are also characters of hypergroups. There are many additional examples from the classical special functions. Indeed, it is possible to give conditions on a Sturm-Liouville problem so that its eigenfunctions must be characters of a hyper group. The converse will also be discussed. If a hypergroup consists of measures on a real (compact or not) interval, then with adequate regularity conditions, it must be the case that the characters of the hypergroup are eigenfunctions of a Sturm-Liouville problem.LECTURE 2. A family of orthogonal polynomials on an interval I may be the characters of a hypergroup of measures supported on I (which would be called a continuous polynomial hypergroup), and the family may also supply the characters of a hypergroup of measures on the discrete set {0, 1, 2, } (which would be called a discrete polynomial hypergroup). The entire category of continuous polynomial hypergroups can be explicitly described, but the full category of discrete polynomial hypergroups has not yet been characterized, though there are some fairly general theorems.LECTURE 3. The same issues in the second talk raise analagous questions for multivariate orthogonal polynomials. A family of multivariate orthogonal polynomials is a much more subtle object than a family of one-variable orthogonal polynomials. Some progress has been made in the classification problem, and these results will be discussed as well as recently discovered examples.KeywordsOrthogonal PolynomialHaar MeasureLegendre PolynomialProduct FormulaMeasure AlgebraThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.