Contractions of Lie algebras: applications to special functions and separation of variables

We investigate the consequences of contraction of the Lie algebras of the o rthogonal groups to the Lie algebras of the Euclidean groups in terms of se paration of variables for Laplace Beltrami eigenvalue equations, and the so lutions of these equations that arise through separation of variables techn iques, on the N-sphere and in N-dimensional Euclidean space. General ellips oidal and paraboloidal coordinates are included, not just the subgroup-type coordinates that have been the concern of most investigations of contracti ons as applied to special functions. We pay special attention to the case N = 2 where we show in detail, for example, how Lame polynomials contract to periodic Mathieu functions. Our point of view emphasizes the characterizat ion of separable polynomial eigenfunctions in terms of the zeros of these e igenfunctions. We also consider all possible separable coordinate systems o n the complex two-sphere (which includes real hyperboloids as special cases ) and their contraction to flat space coordinates.