Symmetric truncated Freud polynomials

We define the family of symmetric truncated Freud polynomials $P_n(x;z)$, orthogonal with respect to the linear functional $u$ defined by $〈u,p\left(x\right)〉={\int }_{-z}^{z}p\left(x\right)e^{-x^4}dx,p\in P,z>0.$The semiclassical character of $P_n(x;z)$ as polynomials of class 4 is stated. As a consequence, several properties of $P_n(x;z)$ concerning the coefficients ${\gamma }_n\left(z\right)$ in the three-term recurrence relation they satisfy, as well as the moments and the Stieltjes function of $u$, are studied. Ladder operators associated with such a linear functional and the holonomic equation satisfied by the polynomials $P_n(x;z)$ are deduced. Additionally, an electrostatic interpretation of their zeros and their dynamics with respect to the parameter $z$ are provided. We also consider a rescaled orthonormal sequence $p_n(x;z)$ supported on the fixed interval $[-1,1]$, with respect to the weight $e^{-z^4{x}^4}$, and establish a relative outer asymptotic relation with the Chebyshev polynomials of the second kind in the complex domain.