@**techreport**{RISC6531,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)}

}