Orthogonal polynomials which are eigenfunctions of a partial differential operator
Abstract
We study orthogonal polynomials of d = d1+d2 variables with respect to a wrapped product weight function W(x,y) = W1(x/ρ(y)) W2(y) for (x, y) ∈ ℝd1 × ℝd2, where ρ is either linear or the square root of a nonnegative quadratic polynomial, and identify all such polynomials that are eigenfunctions of a second-order linear differential operator. For d =2, it is known that there are primarily, up to affine transformations, five families of such polynomials, which are products or wrapped products of classical orthogonal polynomials of one variable; all five families have their counterparts in higher dimensions, but no characterization is known in dimension three or higher. Our study explores viable wrapped product families, finds explicit second-order differential operators for two new families of orthogonal polynomials in d= d1+d2 variables that have not been studied before if either d1>1 or d2 > 1, and provides, in particular, a complete list of such operators among all wrapped product orthogonal polynomials when d = 3. The list also includes four families that are eigenfunctions of a fourth-order differential operator, whereas no second-order operator is available. Moreover, orthogonal polynomials on the wrapped quadratic surfaces that are eigenvalues of a second-order differential operator on the surface are also studied.
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BibTeX
@misc{xu2026orthogonal,
title = {{Orthogonal polynomials which are eigenfunctions of a partial differential operator}},
author = {Yuan Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01751},
eprint = {2609.01751},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01751},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Orthogonal polynomials which are eigenfunctions of a partial differential operator
AU - Yuan Xu
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01751
UR - https://arxiv.org/abs/2609.01751
AB - We study orthogonal polynomials of d = d_1+d_2 variables with respect to a wrapped product weight function W(x,y) = W_1(x/ρ(y)) W_2(y) for (x, y) ∈ ℝ^d_1 × ℝ^d_2, where ρ is either linear or the square root of a nonnegative quadratic polynomial, and identify all such polynomials that are eigenfunctions of a second-order linear differential operator. For d =2, it is known that there are primarily, up to affine transformations, five families of such polynomials, which are products or wrapped products of classical orthogonal polynomials of one variable; all five families have their counterparts in higher dimensions, but no characterization is known in dimension three or higher. Our study explores viable wrapped product families, finds explicit second-order differential operators for two new families of orthogonal polynomials in d= d_1+d_2 variables that have not been studied before if either d_1>1 or d_2 > 1, and provides, in particular, a complete list of such operators among all wrapped product orthogonal polynomials when d = 3. The list also includes four families that are eigenfunctions of a fourth-order differential operator, whereas no second-order operator is available. Moreover, orthogonal polynomials on the wrapped quadratic surfaces that are eigenvalues of a second-order differential operator on the surface are also studied.
ER -