This series presents some tools of applied mathematics in the areas of
proba- bility theory, operator calculus, representation theory, and
special functions used currently, and we expect more and more in the
future, for solving problems in math- ematics, physics, and, now,
computer science. Much of the material is scattered throughout available
literature, however, we have nowhere found in accessible form all of
this material collected. The presentation of the material is original
with the authors. The presentation of probability theory in connection
with group represen- tations is new, this appears in Volume I. Then the
applications to computer science in Volume II are original as well. The
approach found in Volume III, which deals in large part with
infinite-dimensional representations of Lie algebras/Lie groups, is new
as well, being inspired by the desire to find a recursive method for
calcu- lating group representations. One idea behind this is the
possibility of symbolic computation of the matrix elements. In this
volume, Representations and Probability Theory, we present an intro-
duction to Lie algebras and Lie groups emphasizing the connections with
operator calculus, which we interpret through representations,
principally, the action of the Lie algebras on spaces of polynomials.
The main features are the connection with probability theory via moment
systems and the connection with the classical ele- mentary distributions
via representation theory. The various systems of polynomi- als that
arise are one of the most interesting aspects of this study.