  Let $X=\{X_t, t\ge0\}$ be a c\`{a}dl\`{a}g L\'{e}vy process, centered, with
moments of all orders. There are two families of orthogonal polynomials
associated with $X$. On one hand, the Kailath--Segall formula gives the
relationship between the iterated integrals and the variations of order $n$ of
$X$, and defines a family of polynomials $P_1(x_1), P_2(x_1,x_2),...$ that are
orthogonal with respect to the joint law of the variations of $X$. On the other
hand, we can construct a sequence of orthogonal polynomials $p^{\sigma}_n(x)$
with respect to the measure $\sigma^2\delta_0(dx)+x^2 \nu(dx)$, where
$\sigma^2$ is the variance of the Gaussian part of $X$ and $\nu$ its L\'{e}vy
measure. These polynomials are the building blocks of a kind of chaotic
representation of the square functionals of the L\'{e}vy process proved by
Nualart and Schoutens. The main objective of this work is to study the
probabilistic properties and the relationship of the two families of
polynomials. In particular, the L\'{e}vy processes such that the associated
polynomials $P_n(x_1,...,x_n)$ depend on a fixed number of variables are
characterized. Also, we give a sequence of L\'{e}vy processes that converge in
the Skorohod topology to $X$, such that all variations and iterated integrals
of the sequence converge to the variations and iterated integrals of $X$.
