Symmetrization process and truncated orthogonal polynomials

We define the family of truncated Laguerre polynomials \(P_n(x;z)\) $P_n(x;z)$ , orthogonal with respect to the linear functional \(\varvec{\ell }\) $\mathit{\ell }$ defined by \(\begin{aligned} \left\langle {\varvec{\ell },p}\right\rangle =\int _{0}^zp(x)x^\alpha e^{-x}dx,\qquad \alpha >-1. \end{aligned}\) $$\begin{array}{c}\hfill 〈\mathit{\ell },p〉={\int }_0^{z}p(x)x^{\alpha }{e}^{-x}dx,\alpha >-1.\end{array}$$ The connection between \(P_n(x;z)\) $P_n(x;z)$ and the polynomials \(S_n(x;z)\) $S_n(x;z)$ (obtained through the symmetrization process) constitutes a key element in our analysis. As a consequence, several properties of the polynomials \(P_n(x;z)\) $P_n(x;z)$ and \(S_n(x;z)\) $S_n(x;z)$ 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.