Symmetric Truncated Freud polynomials
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arXiv
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| Format: | Preprint |
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2024
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| author | Huertas, Edmundo J. Lastra, Alberto Marcellán, Francisco Soto-Larrosa, Víctor |
| author_facet | Huertas, Edmundo J. Lastra, Alberto Marcellán, Francisco Soto-Larrosa, Víctor |
| contents | 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 $γ_n (z)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_00764 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Symmetric Truncated Freud polynomials Huertas, Edmundo J. Lastra, Alberto Marcellán, Francisco Soto-Larrosa, Víctor Classical Analysis and ODEs 33C45, 33C47 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 $γ_n (z)$ 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. |
| title | Symmetric Truncated Freud polynomials |
| topic | Classical Analysis and ODEs 33C45, 33C47 |
| url | https://arxiv.org/abs/2412.00764 |