  We define the family of truncated Laguerre polynomials $P_n(x;z)$, orthogonal
with respect to the linear functional $\ell$ defined by
$$\langle{\ell,p\rangle}=\int_{0}^zp(x)x^\alpha e^{-x}dx,\qquad\alpha>-1.$$
  The connection between $P_n(x;z)$ and the polynomials $S_n(x;z)$ (obtained
through the symmetrization process) constitutes a key element in our analysis.
As a consequence, several properties of the polynomials $P_n(x;z)$ and
$S_n(x;z)$ are studied taking into account the relation between the parameters
of the three-term recurrence relations that they satisfy. Asymptotic expansions
of these coefficients are given. Discrete Painlev\'e and Painlev\'e equations
associated with such coefficients appear in a natural way. An electrostatic
interpretation of the zeros of such polynomials as well as the dynamics of the
zeros in terms of the parameter $z$ are given.
