Symmetrization process and truncated orthogonal polynomials

We define the family of truncated Laguerre polynomials Pn(x; z), orthogonal with respect to the linear functional t defined by
(t, p ) =z0p(x)xαe−xdx, α> −1.
The connection between Pn(x; z) and the polynomials Sn(x; z) (obtained through the symmetrization process) constitutes a key element in our analysis. As a consequence, several properties of the polynomials Pn(x; z) and Sn(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.

truncated laguerre polynomials; symmetrization process; pearson equation; laguerre–freud equations; ladder operators; painlevé equations; zeros; 42c05; 33c50

Symmetrization process and truncated orthogonal polynomials Articles