Spherical Functions on Euclidean Space

We study special functions on euclidean spaces from the viewpoint of riemannian symmetric spaces. Here the euclidean space $E^n=G\mathrm{/}K$ where $G$ is the semidirect product $R^n\cdot K$ of the translation group with a closed subgroup $K$ of the orthogonal group O(n). We give exact parameterizations of the space of $(G,K)$–spherical functions by a certain affine algebraic variety, and of the positive definite ones by a real form of that variety. We give exact formulae for the spherical functions in the case where $K$ is transitive on the unit sphere in $E^n$.