  It is well known that between all processes with independent increments,
essentially only the Brownian motion and the Poisson process possess the
chaotic representation property (CRP). Thus, a natural question appears: What
is an appropriate analog of the CRP in the case of a general L\'evy process. At
least three approaches are possible here. The first one, due to
  It\^o, uses the CRP of the Brownian motion and the Poisson process, as well
as the representation of a L\'evy process through those processes. The second
approach, due to Nualart and Schoutens, consists in representing any
square-integrable random variable as a sum of multiple stochastic integrals
constructed with respect to a family of orthogonalized centered power jumps
processes. The third approach, never applied before to the L\'evy processes,
uses the idea of orthogonalization of polynomials with respect to a probability
measure defined on the dual of a nuclear space. The main aims of the present
paper are to develop the three approaches in the case of a general
($\R$-valued) L\'evy process on a Riemannian manifold and (what is more
important) to understand a relationship between these approaches. We apply the
obtained results to the gamma, Pascal, and Meixner processes, in which case the
analysis related to the orthogonalized polynomials becomes essentially simpler
and richer than in the general case.
