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 \(\varvec{\ell }\) defined by

$$\begin{aligned} \left\langle {\varvec{\ell },p}\right\rangle =\int _{0}^zp(x)x^\alpha e^{-x}dx,\qquad \alpha >-1. \end{aligned}$$

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 Painlevé and Painlevé 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.