Lie Theory and Separation of Variables. II: Parabolic Coordinates

Winternitz and coworkers have characterized those solutions of the equation $(\Delta _3 + \omega ^2) f(x) = 0$ which are expressible as products of functions of the paraboloid of revolution, as simultaneous eigenfunctions of the commuting quadratic operators \[E = J_1 P_2 , + P_2 J_1 , - P_1 J_2 - J_2 P_1 ,J_3^2 \] in the enveloping algebra of the Lie algebra of the Euclidean group in three-space $E(3)$. Here we study the representation theory of the real and complex Euclidean groups in an $E - J_3^2 $ basis and use the results to derive some addition and expansion theorems for parabolic functions which simplify and in some cases extend identities due to Buchholz and Hochstadt. We also give the decomposition of the quasi-regular representation of $E(3)$ in an $E - J_3^2 $ basis.