SEPARATION OF VARIABLES AND LIE-ALGEBRA CONTRACTIONS.
APPLICATIONS TO SPECIAL FUNCTIONS

G. Pogosyan
Centro de Ciencias F\'{\i}sicas
Universidad Nacional Aut\'onoma de M\'exico, Morelos, M\'exico

Joint Institute for Nuclear Research, Dubna

International Center for Advanced Studies,
Yerevan State University, Yerevan, Armenia

A. Sissakian
  Joint Institute for Nuclear Research, Dubna
P. Winternitz
Centre de recherches math{\'e}matiques,
Universit{\'e} de Montr{\'e}al, Montr{\'e}al, Qu{\'e}bec, Canada

A review is given of some recently obtained results on analytic
contractions of Lie algebras and Lie groups and their application to
special function theory. The contractions considered are from $O(3)$
to $E(2)$ and from $O(2,1)$ to $E(2)$, or $E(1,1)$. The analytic
contractions provide relations between separable coordinate systems
on various homogeneous manifolds. They lead to asymptotic relations
between basis functions and overlap functions for the representations
of different groups.