author = {Diego Dominici and Francisco Marcell{\'a}n},
title = {{Truncated Hermite polynomials}},
language = {english},
abstract = {We define the family of truncated Hermite polynomials $P_{n}left( x;zright) $, orthogonal with respect to the linear functional [Lleft[ pright] = int_{-z}^{z} pleft( xright) e^{-x^{2}} ,dx. ] The connection of $P_{n}left( x;zright) $ with the Hermite and Rys polynomials is stated. The semiclassical character of $P_{n}left( x;zright) $ as polynomials of class $2$ is emphasized. As a consequence, several properties of $P_{n}left( x;zright) $ concerning the coefficients $gamma_{n}left( zright) $ in the three-term recurrence relation they satisfy as well as the moments and the Stieltjes function of $L$ are given. Ladder operators associated with the linear functional $L$, a holonomic differential equation (in $x)$ for the polynomials $P_{n}left( x;zright) $, and a nonlinear ODE for the functions $gamma_{n}left( zright) $ are deduced. },
number = {22-10},
year = {2022},
month = {August},
keywords = {Orthogonal polynomials, Gaussian distribution},
length = {37},
license = {CC BY 4.0 International},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger Straße 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}