Symmetrization process and truncated orthogonal polynomials

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We define the family of truncated Laguerre polynomials Pn(x;z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_n(x;z)$$\\end{document}, orthogonal with respect to the linear functional ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\\ell }$$\\end{document} defined by ℓ,p=∫0zp(x)xαe-xdx,α>-1.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\langle {\\varvec{\\ell },p}\\right\\rangle =\\int _{0}^zp(x)x^\\alpha e^{-x}dx,\\qquad \\alpha >-1. \\end{aligned}$$\\end{document}The connection between Pn(x;z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_n(x;z)$$\\end{document} and the polynomials Sn(x;z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_n(x;z)$$\\end{document} (obtained through the symmetrization process) constitutes a key element in our analysis. As a consequence, several properties of the polynomials Pn(x;z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_n(x;z)$$\\end{document} and Sn(x;z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_n(x;z)$$\\end{document} 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.