Contractions of lie algebras: Applications to special functions and separation of variables

We investigate the consequences of contraction of the Lie algebras of the orthogonal groups to the Lie algebras of the Euclidean groups in terms of separation of variables for Laplace-Beltrami eigenvalue equations, and the solutions of these equations that arise through separation of variables techniques, on the N-sphere and in N-dimensional Euclidean space. General ellipsoidal and paraboloidal coordinates are included, not just the subgroup-type coordinates that have been the concern of most investigations of contractions as applied to special functions. We pay special attention to the case N = 2 where we show in detail, for example, how Lamé polynomials contract to periodic Mathieu functions. Our point of view emphasizes the characterization of separable polynomial eigenfunctions in terms of the zeros of these eigenfunctions. We also consider all possible separable coordinate systems on the complex two-sphere (which includes real hyperboloids as special cases) and their contraction to flat space coordinates.