Symmetrization process and truncated orthogonal polynomials

We define the family of truncated Laguerre polynomials $P_n(x;z)$, orthogonal with respect to the linear functional $elln$ such that $$prodint{elln,p}=int_{0}^zp(x)x^alpha e^{-x}dx,qquadalpha>-1.$$ 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 such 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 such zeros in terms of the parameter $z$ are given.