Continuous Hahn Polynomials of Differential Operator Argument and Analysis onRiemannian Symmetric Spaces of Constant Curvature

摘要

The phenomenon associating spherical functions on Euclidean space and on spacesof constant positive and negative curvature is addressed. For the three types of simply connected Riemannian spaces of constant curvature it is shown that the associated spherical functions can be obtained from the corresponding (zonal) spherical functions by application of a differential operator of the form p(i d/dt), where p belongs to a system of orthogonal polynomials: Gegenbauer, Hahn or continuous symmetric Hahn polynomials. A group theoretic explanation of this phenomenon is given and the properties of the polynomials p are related to the properties of the corresponding representation. The method is extended to the case of intertwining functions.