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arXiv · 2412.00764

Symmetric Truncated Freud polynomials

Edmundo J. Huertas·Alberto Lastra·Francisco Marcellán·Víctor Soto-Larrosa
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Abstract

We define the family of symmetric truncated Freud polynomials $P_n(x;z)$, orthogonal with respect to the linear functional $\mathbf{u}$ defined by \begin{equation*} \langle \mathbf{u}, p(x)\rangle = \int_{-z}^z p(x)e^{-x^4}dx, \quad p\in \mathbb{P}, \quad z>0. \end{equation*} 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 $\gamma_n (z)$ in the three-term recurrence relation they satisfy as well as the moments and the Stieltjes function of $\mathbf{u}$ are studied. Ladder operators associated with such a linear functional and the holonomic equation that the polynomials $P_n (x; z)$ satisfy are deduced. Finally, an electrostatic interpretation of the zeros of such polynomials and their dynamics in terms of the parameter $z$ are given.

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Edmundo J. Huertas, Alberto Lastra, Francisco Marcellán, Víctor Soto-Larrosa. 2024-12-01. Symmetric Truncated Freud polynomials. https://arxiv.org/abs/2412.00764

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